Your Construction 1. is very similar to this: Let $R_{\mathsf{I\Sigma_2}}$ be the following fast-growing function $$\begin{aligned} R_{\mathsf{I\Sigma_2}}\colon x \longmapsto \sup \{ y\mid & y\mbox{ is the least witness for a $\Sigma_1$ sentence provable in } \\ & \mbox{$\mathsf{I\Sigma_2}$ by a proof with the Gödel number $\le x$}\}\end{aligned}.$$ $T_i$'s from the example by Walsh and me could be equivalently given as $$T_i\colon \mathsf{EA}+\forall x\; (R_{\mathsf{I}\Sigma_2}(x+i)\mbox{ is defined } \Rightarrow \mathsf{RFN}^x_{\Sigma_1}(\mathsf{EA}))+R_{\mathsf{I}\Sigma_2}\mbox{ isn't total}.$$

As for your last conjecture. It is indeed the case that for sequences of sound arithmetical theories $T_i's$ it couldn't be the case that some $T_i\vdash \forall x\;(\mathsf{Prv}_{T_x}(\mathsf{Con}(T_{x+1})))$ (this is due to Friedman, Smorynski, and Solovay; in my paper with Walsh that I mentioned earlier we give a simple proof of this (Theorem 3.6); you could find reference to the earlier works about it there just before Theorem 3.6).

In the construction above one could replace $\mathsf{EA}$ with $\mathsf{I}\Sigma_1$ and sustain the same property. But in the case of $\mathsf{I}\Sigma_1$ the $T_i$'s could be defined in an alternative way that makes them very similar to your Construction 1. Let $R_{\mathsf{I\Sigma_2}}$ be the following fast-growing function $$\begin{aligned} R_{\mathsf{I\Sigma_2}}\colon x \longmapsto \sup \{ y\mid & y\mbox{ is the least witness for a $\Sigma_1$ sentence provable in } \\ & \mbox{$\mathsf{I\Sigma_2}$ by a proof with the Gödel number $\le x$}\}\end{aligned}.$$ Alternatively $T_i$'s could be defined as $$T_i\colon \mathsf{I}\Sigma_1+\forall x\; (R_{\mathsf{I}\Sigma_2}(x+i)\mbox{ is defined } \Rightarrow \mathsf{RFN}^x_{\Sigma_1}(\mathsf{I}\Sigma_1))+R_{\mathsf{I}\Sigma_2}\mbox{ isn't total}.$$

As for your last conjecture. It is known that for any sound arithmetical theory $U$ and a recursive chain $T_i$ of r.e. extensions of $U$ it couldn't be the case that $U\vdash \forall x\;(\mathsf{Prv}_{T_x}(\mathsf{Con}(T_{x+1})))$ (this is due to Friedman, Smorynski, and Solovay; in my paper with Walsh that I mentioned earlier we give a simple proof of this (Theorem 3.6); you could find reference to the earlier works about it there just before Theorem 3.6).

A more natural example of a descending chain that I know is based on the notion of slow consistency. For a computable function $f$ let $$\mathsf{PA}\upharpoonright f =\{\mathsf{I}\Sigma_x\mid f(x) \mbox{ is defined}\}.$$ In the case when $f$ is total but not provably total in some theory $T$ the theory $\mathsf{PA}\upharpoonright f$ from the external point of view just coincide with $\mathsf{PA}$, but $T$ might not be able to prove that $\mathsf{PA}$ and $\mathsf{PA}\upharpoonright f$ coincide. For rationals $q>0$ let me consider the functions $f_{q}(x)=\mathsf{h}_{\varepsilon_0}([qx])$. Let $T_q=\mathsf{I}\Sigma_1+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$. The sentences $\mathsf{Con}(\mathsf{PA}\upharpoonright f)$ are known as slow consistency sentences and were introduced by Friedman, Rathjen, and Weiermann. The claim is that for $p>q>0$ we have $T_q\vdash \mathsf{Con}(T_p)$. Let me reason in $T_q$. Actually I will present a model-theoretic argument that isn't directly available in an extension of $\mathsf{I\Sigma}_1$, but with some additional care it could be formalized in $\mathsf{WKL}_0+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$ and then transfered to $T_q$ by the arithmetical conservativity of $\mathsf{WKL}_0$ over $\mathsf{I}\Sigma_1$. First assume that $f_q$ is total, this actually is equivalent to $\Sigma_1$-soundness of $\mathsf{PA}$, and it is fairly easy to prove consistency of $T_p$ in this case. Now we assume that there is the number $a$ s.t. $f_q(a)$ is defined but $f_q(a+1)$ is not. Since we externally know that $f_q$ is total, for any given $N$ we could prove in $T_q$ that $a>N$. By taking $N$ to be large enough we prove in $T_q$ that there is a number $b$ s.t. $f_p(b)$ is undefined and $[bp]<[ap]$. Clearly we have $\mathsf{PA}\upharpoonright f_q=\mathsf{I}\Sigma_a$. We construct a model $\mathfrak{M}$ of $\mathsf{I}\Sigma_a$. If $f_p(a)$ is undefined in $\mathfrak{M}$, then we are done since $\mathfrak{M}\models \mathsf{PA}\upharpoonright f_p\subseteq \mathsf{I}\Sigma_{a-1}$, $\mathsf{I}\Sigma_a\vdash \mathsf{Con}(\mathsf{I}\Sigma_{a-1})$ and hence $\mathfrak{M}\models \mathsf{Con}( \mathsf{PA}\upharpoonright f_p)$. Otherwise, in $\mathfrak{M}$ we have a non-standard interval $[f_p(b),f_p(a)]$. Since for any tower of $2$-exponentiation we have $$\mathfrak{M}\models 2^{\displaystyle \ldots^{\displaystyle 2^{\displaystyle f_p(b)}}}<f_p(a),$$ there is a cut $\mathfrak{J}$ of $\mathfrak{M}$ s.t. $f_p(b)<\mathfrak{J}<f_p(a)$ and $\mathfrak{J}\models \mathsf{EA}$. Clearly, $$\mathfrak{J}\models \mathsf{Con}(\mathsf{I}\Sigma_{a-1})\;\;\;\mbox{ and }\;\;\;\mathfrak{J}\models f_p(a)\mbox{ is undefined}$$ and hence $\mathfrak{J}\models \mathsf{Con}(\mathsf{PA}\upharpoonright f_p)$.

A more natural example of a descending chain that I know is based on the notion of slow consistency. For a computable function $f$ let $$\mathsf{PA}\upharpoonright f =\{\mathsf{I}\Sigma_x\mid f(x) \mbox{ is defined}\}.$$ In the case when $f$ is total but not provably total in some theory $T$ the theory $\mathsf{PA}\upharpoonright f$ from the external point of view just coincide with $\mathsf{PA}$, but $T$ might not be able to prove that $\mathsf{PA}$ and $\mathsf{PA}\upharpoonright f$ coincide. The sentences $\mathsf{Con}(\mathsf{PA}\upharpoonright f)$ are known as slow consistency sentences and were introduced by Friedman, Rathjen, and Weiermann. For rationals $q>0$ let me consider the functions $f_{q}(x)=\mathsf{h}_{\varepsilon_0}([qx])$. Let $T_q=\mathsf{I}\Sigma_1+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$. The claim is that for $p>q>0$ we have $T_q\vdash \mathsf{Con}(T_p)$. Let me reason in $T_q$. Actually I will present a model-theoretic argument that isn't directly available in an extension of $\mathsf{I\Sigma}_1$, but with some additional care it could be formalized in $\mathsf{WKL}_0+\mathsf{Con}(\mathsf{PA}\upharpoonright f_q)$ and then transfered to $T_q$ by the arithmetical conservativity of $\mathsf{WKL}_0$ over $\mathsf{I}\Sigma_1$. First assume that $f_q$ is total, this actually is equivalent to $\Sigma_1$-soundness of $\mathsf{PA}$, and it is fairly easy to prove consistency of $T_p$ in this case. Now we assume that there is the number $a$ s.t. $f_q(a)$ is defined but $f_q(a+1)$ is not. Since we externally know that $f_q$ is total, for any given $N$ we could prove in $T_q$ that $a>N$. By taking $N$ to be large enough we prove in $T_q$ that there is a number $b$ s.t. $f_p(b)$ is undefined and $[bp]<[ap]$. Clearly we have $\mathsf{PA}\upharpoonright f_q=\mathsf{I}\Sigma_a$. We construct a model $\mathfrak{M}$ of $\mathsf{I}\Sigma_a$. If $f_p(a)$ is undefined in $\mathfrak{M}$, then we are done since $\mathfrak{M}\models \mathsf{PA}\upharpoonright f_p\subseteq \mathsf{I}\Sigma_{a-1}$, $\mathsf{I}\Sigma_a\vdash \mathsf{Con}(\mathsf{I}\Sigma_{a-1})$ and hence $\mathfrak{M}\models \mathsf{Con}( \mathsf{PA}\upharpoonright f_p)$. Otherwise, in $\mathfrak{M}$ we have a non-standard interval $[f_p(b),f_p(a)]$. Using the fact that $[pb]<[pa]$ it is easy to show that $\mathfrak{M}\models A(f_p(b))<f_p(a)$, where $A(x)$ is the diagonal of Ackermann's function. From Sommer's results it follow that there is a cut $\mathfrak{J}\models \mathsf{I}\Sigma_1$ of $\mathfrak{M}$ such that $f_p(b)<\mathfrak{J}<f_p(a)$. Clearly we have $$\mathfrak{J}\models \mathsf{Con}(\mathsf{I}\Sigma_{a-1})\;\;\;\mbox{ and }\;\;\;\mathfrak{J}\models f_p(a)\mbox{ is undefined}$$ and hence $\mathfrak{J}\models \mathsf{Con}(\mathsf{PA}\upharpoonright f_p)$.