* what are some methods for removing the water? I tried sweeping some
directly into the gutter downspout, but that takes a LONG time with
very little effect and doesn't work for all the puddles given where
the puddles are and where the downspouts are?

Been there, done that. Get one or more sump pumps, discharge hose(s)
through downspout. Be watchful that the hoses don't eventually start to
compress where they bend, though. Some can run with minimal water depth;
the ones that have an "automatic on" water sensing feature tend to need
more water depth in order for the float to trigger the operation, so I
found that the cheaper "no float" ones worked better in the long run.
For this reason Home Depot's variety didn't work as well as the "Ace" brand.

Does your house have excessive humidity (e.g. windows fog, condensation
on sills)? if so, the roof is undoubtedly in bad shape, letting in
excess moisture in addition to any obvious water leakage.

Good luck.