Citrus always sets way more fruit than it actually holds. It is normal for most of the fruit to fall off like you decribe. How large are the trees? If they are too small they may not hold any fruit at all. Not sure on the dieing branches issue there are tons of possible causes. What type of potting mix are they in? Meyer lemons are actually kind of tough to keep happy in pots.

They are in a mix of mostly citrus/cactus storebought media, some perlite, and a little bit of standard potting soil (ran out of the other two components). Both plants are 20" x 20" or so.

to clarify, they were only repotted into slightly bigger pots after purchase, so mostly, they're in whatever media Monrovia grew them in.

As I said, the trees came from the store with half-formed green lemons on it already. I cut all the fruit off of one of the trees, and left maybe 2/3 of the fruit (maybe 10 lemons), on the other one. Both seem very healthy, but the tree that had all original fruit cut off seems to be growing new leaves faster. However, it would be nice if the half-formed fruit on the other tree could finish maturing and I could see some lemons. So, a question I have is, should I keep the fruit that originally came on the tree (I bought it maybe 3 months ago and the fruit has remained, but has not gotten bigger) on the one tree and see what happens, or should I get rid of it?