First, I am assuming below that the vector of variables (e.g., the vector $(A,B)$) follows a multivariate normal distribution. This is more than assuming that they are marginally normal, and needed to solve your problems.
Problem 1: Remark that $B-A\sim\mathcal{N}(\mu_b-\mu_a,\sigma^2_a+\sigma^2_b)$. Then,
$$\Pr(A>B)=\Pr(B-A<0)=\Phi\left(\frac{\mu_b-\mu_a}{\sigma^2_a+\sigma^2_b}\right),$$
where $\Phi$ is the cumulative distribution function of a standard normal distribution $\mathcal{N}(0,1)$ (see here for details).
Problem 2: For simplicity I rather denote the variables by $(X_1,…,X_n)$, with $X_i\sim \mathcal{N}(\mu_i,\sigma^2_i)$. I also let $\varphi$ denote the density of a standard normal. Then,
\begin{align*}
\Pr(X_1=\max(X_1,…,X_n)) &= \Pr(X_1>X_2,…,X_1>X_n) \\
& = E\left[\Pr(X_1>X_2,…,X_1>X_n|X_1)\right] \\
& = \int \Pr(x_1>X_2,…,x_1>X_n) \frac{1}{\sigma_1}\varphi\left(\frac{x-\mu_1}{\sigma_1}\right)dx_1\\
& = \int \prod_{i=2}^n \Phi\left(\frac{x_1-\mu_i}{\sigma_i}\right) \frac{1}{\sigma_1}\varphi\left(\frac{x-\mu_1}{\sigma_1}\right)dx_1.
\end{align*}
(The third equality follows by independence of the different $X_i)
I don't think there is a closed-form expression for this integral, but it can be accurately and quickly approximated by a Gauss-Hermite quadrature, see here.
Problem 3: In this case, let $Y_i=X_1 - X_{i+1}$ for $i=1,…,n-1$. Then $(Y_1,…,Y_{n-1})$ follows a multivariate normal distribution and we seek to compute $\Pr(Y_1>0,…,Y_{n-1}>0)$. This is a particular instance of computing that a normal random vector $Y$ belongs to a hyperrectangle $A$ (say). There are several methods for approximating this probability, but the most popular one is probably the GHK algorithm. Intuitively and compared to "naive" Monte Carlo simulations, the idea is to draw variables so that they are more informative about the event $\{Y\in A\}$.
Best Answer
I cannot comment (not enough reputation).
Vincent: You have the wrong pdf for $g(x)$, you have a normal distribution with mean 1 and variance 1, not mean $\mu$.
Hint: You don't need to solve any integrals. You should be able to write this as pdf's and their expected values, so you never need to integrate.
Outline: Firstly, $ \log({f(x) \over g(x) })=\left\{ -{1 \over 2} \left( x^2 - (x-\mu )^2 \right) \right\} $ . Expand and simplify. Don't even write out the other $f(x)$ and see where that takes you.