In answer to (1):
Set
$\theta_i \sim \text{Beta}(\alpha, \beta)$
Then the marginal data distribution, $\int_0^1\Pr(y_i|\theta_i)p(\theta_i|\alpha, \beta)d\theta_i$, is a standard distribution:
$y_i \sim \text{Beta-binomial}(\alpha, \beta)$
(Wikipedia article: Beta-binomial distribution; see especially the section on "Further Bayesian considerations".) Since this results in a bivariate posterior distribution, you can easily evaluate it on a finely-spaced grid; thus MCMC can be avoided for any choice of hyperpriors.
More specifically, you can find the region of high posterior density by numerical optimization, using the ML estimate as the initial guess. After finding the region containing all but a negligible amount of posterior probability mass and evaluating the posterior density on the fine grid, you can generate random $(\alpha, \beta)$ pairs by picking one of the points on the grid with probability proportional to the evaluated density and then adding uniform jitter centered at the selected point with width equal to the grid spacing.
If you're interested in inferring the $\theta_i$ variables, you can sample them by first sampling a set of $(\alpha, \beta)$ values from their posterior and then sampling the $\theta_i$ from their conditional posteriors,
$\theta_i|y_i, \alpha, \beta \sim \text{Beta}(\alpha + y_i, \beta + n_i - y_i)$