let me work out the comments a bit further, starting from the identity (equation 8.2 from <A HREF="http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf">Diaconis and Evans</A>, correcting an earlier <A HREF="http://statweb.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf">paper</A> by Diaconis and Shahshahani)

$$\int_{{\rm O}(n)}{\rm tr}\,g^p\,dg=\begin{cases}
0&{\rm if}\;\;p\;\;\text{is an odd integer}\\
1&{\rm if}\;\;p\;\;\text{is an even integer}
\end{cases}
$$

so the second integral of the OP evaluates to
$$\int_{{\rm O}(n)}g^p\,dg=\begin{cases}
0&{\rm if}\;\;p\;\;\text{is an odd integer}\\
n^{-1}\mathbb{1}&{\rm if}\;\;p\;\;\text{is an even integer}
\end{cases}
$$

Taylor expansion of the exponent in the first integral of the OP and term-by-term integration gives
$$
\int_{{\rm O}(n)}e^g\,dg=n^{-1}\mathbb{1}\sum_{p=0}^\infty \frac{1}{(2p)!}=\frac{\cosh 1}{n}\mathbb{1}
$$

now for the third integral of the OP, we need the fourth-order tensor
$$\int_{{\rm O}(n)}(g^p)_{ij}(g^q)_{kl}\,dg=a_{pq}(n)\delta_{ij}\delta_{kl}+b_{pq}(n)\delta_{ik}\delta_{jl}+c_{pq}(n)\delta_{il}\delta_{jk}$$
so that the required integral takes the form
$$\int_{{\rm O}(n)}g^p Ag^q\,dg=a_{pq}(n)A+b_{pq}(n)A^{\rm t}+c_{pq}(n)\mathbb{1}\,{\rm tr}\,A
$$
by taking traces I find, using again equation 8.2 from <A HREF="http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf">Diaconis and Evans</A>, that
$$na_{pq}(n)+nb_{pq}(n)+n^2 c_{pq}(n)=\int_{{\rm O}(n)}{\rm tr}\,g^{p+q}\,dg=\begin{cases}
0&{\rm if}\;\;p+q\;\;\text{is an odd integer}\\
1&{\rm if}\;\;p+q\;\;\text{is an even integer}
\end{cases}
$$
$$n^2a_{pq}(n)+nb_{pq}(n)+nc_{pq}(n)=\int_{{\rm O}(n)}({\rm tr}\,g^p)({\rm tr}\,g^q)\,dg=\begin{cases}
{\rm min}\,(p,2n)&{\rm if}\;\;p=q\;\;{\rm odd}\\
{\rm min}\,(p,2n)+1&{\rm if}\;\;p=q\;\;{\rm even}\\
1&{\rm if}\;\;p\neq q\;\;\text{both even}\\
0&{\rm otherwise}
\end{cases}
$$
$$na_{pq}(n)+n^2 b_{pq}(n)+nc_{pq}(n)=\int_{{\rm O}(n)}{\rm tr}\,g^{|p-q|}\,dg=\begin{cases}
0&{\rm if}\;\;p+q\;\;\text{is an odd integer}\\
1&{\rm if}\;\;p+q\;\;\text{is an even integer}
\end{cases}
$$
Three equations with three unknowns solves the third integral.