I doubt there's going to be a simple closed form for the distribution of the sum of inverse gamma variables in general (there might be some special cases that work out); you may be able to show it's equal to some series.
The ratio is something we can say more about.
If $X$ and $Y$ are inverse gamma, and $X_1=1/X$, $Y_1=1/Y$, then $X/Y = Y_1/X_1$, where the two subscripted variates are gamma distributed.
When their scale parameters are identical as here, that ratio is distributed as beta-prime (sometimes called a 'beta distribution of the second kind'). A simple rescaling of that produces an F distribution, so it's also a scaled F, but since the shape parameters are identical here, the scale factor should be 1.
Which is to say, if you have both shape parameters being $\alpha$ in the original inverse gamma distributions, you should get an $F(2\alpha,2\alpha)$ for the ratio.
[If the scale parameters are not identical, you can take the ratio of their scale parameters out as a multiplicative constant, and leave two gammas with identical scales -- so the ratio then has a scaled beta-prime distribution; again also a scaled F, though in this case the scale factor will be different from the one for the beta prime.]
Now that I've looked, I'm astonished that the relation to both the beta prime and the F is not listed on at least the Gamma distribution page. Both pieces of information are on the beta prime page.