For $t\in(-1,1)$, let 
$$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$
and 
$$g(t):=\frac1{f(t)}.$$
Note that the functions $f$ and $g$ are even. 

**Question 1:** Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$? 

**Question 2:** Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive? 

**Question 3:** Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$? 

*Comment:* It appears that 
$$\ln(2g(t))=\sum_{k=1}^\infty\frac{t^{2k}}{2n(2n-1)}.$$ 
If this is true, then the positive answer to 

A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.