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Statement (a) is true for $N$ sufficiently large, and current technology is probably capable of showing it for all $N$'s. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost every even number is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but proving this for almost every even integer seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.

Added. As Jan-Christoph Schlage-Puchta explained in his response, statement (a) is true for every $N$. In fact the original estimates for Goldbach exceptions, due to van der Corput (1937), Tchudakoff (1938), Estermann (1938), suffice for his argument.

Statement (a) is true for $N$ sufficiently large, and current technology is probably capable of showing it for all $N$'s. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost every even number is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but proving this for almost every even integer seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.

Added. As Jan-Christoph Schlage-Puchta explained in his response, statement (a) is true for every $N$. In fact the original estimates for Goldbach exceptions, due to van der Corput (1937), Tchudakoff (1938), Estermann (1938), suffice for his argument.

Statement (a) is true for $N$ sufficiently large, and current technology is probably capable of showing it for all $N$'s. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost every even number is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but proving this for almost every even integer seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.

Statement (a) is true for $N$ sufficiently large, and current technology is probably capable of showing it for all $N$'s. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost every even number is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but proving this for almost every even integer seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.

Added. As Jan-Christoph Schlage-Puchta explained in his response, statement (a) is true for every $N$. In fact the original estimates for Goldbach exceptions, due to van der Corput (1937), Tchudakoff (1938), Estermann (1938), suffice for his argument.

Statement (a) is true for $N$ sufficiently large. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost all even numbers (namely all even numbers but the numbers $p-9$) is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but almost all even integers seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.

Statement (a) is true for $N$ sufficiently large, and current technology is probably capable of showing it for all $N$'s. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number can be written as $p_1+p_2-p_3$ with primes $p_1,p_2,p_3$, and this was discussed in this earlier MO post.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not a difference of two primes). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost every even number is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the lower density of such integers exceeds $1/354$ as proved here), but proving this for almost every even integer seems to be out of reach. On the other hand, assuming a generalized Elliott-Halberstam conjecture, statement (b) follows for any even integers $0<b<a$ such that $a\equiv 0\pmod{3}$ or $b\equiv 0\pmod{3}$ or $a\equiv b\pmod{3}$. Specifically, under these hypotheses, Polymath8b proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes.

To summarize, (a) is essentially known, while (b) seems to be out of reach.