"There are also a bunch of small black dots that look like droppings of some kind."

Likely worm droppings. [See link]

Sometimes hornworms (or other tomato worms) are very hard to spot.

"I'm not clear on how much I'm supposed to water, so I water every morning, maybe 3-4 cups of water. There are also some leaves that are just turning yellow, without any of the holes."

3-4 cups of water/day sounds like a lot for a roma (determinate) for the climate you are in, even for a container plant. Get some light colored mulch (preferably straw) and apply generously. Also, the soil surface should not be constantly moist. Allow the top couple of inches to dry out before watering. Stick your finger in the soil to tell.

Overwatering can contribute to BER, or underwatering for that matter (uneven moisture levels). BER is more common in container plants, and also more common in roma types. It tends to diminish as the season progresses and the plant becomes more mature.

Here is a link that might be useful: hornworm droppings

I'm sure you're right about it being a worm of some type. I've looked very carefully (during the day) and still can't spot anything. My cucumber leaves are having the same problems with the same droppings and I don't see any worms on them either. Without knowing what the worm actually is - is there anything I can spray to get rid of these pests before they strip my plants?
Thanks!

Bt (Bacillius thuringiensis), either in liquid or powdered form. Worm eats piece of leaf coated with Bt, worm gets sick, stops eating, then dies shortly thereafter.

Common brands include Dipel (dust), Greenlight Bt worm killer, and Thuricide. You should be able to pick some up in any gardening center.

Just so you know, it's not a one time spray (it washes off and degrades over time in sunlight). You'll need to apply it once a week or as directions indicate.

Here is a link that might be useful: what are the common tomato pests? -- FAQ

Ok, addressing this part:
"My cucumber leaves are having the same problems with the same droppings and I don't see any worms on them either."

Well, I know next to nothing about cucumbers, but I don't think hormworms mess with them.

Cucumber beetles and just beetles in general leave visible droppings, I believe. So, you might have more than one pest.

Sevin would work on both your tomatoes or cucumbers for eliminating both worms and beetles, but it is pretty broad spectrum, so I don't use it all that much.

Your call.

Hi stephrhuff - I have the same problem you do, but not quite as bad - yet! My container tomato plant on my balcony has the same little black specks on the leaves. They look like black sesame seeds, some strung together by a silk-like string. I just noticed them a few weeks ago and now they're starting to spread. Since I'm not a big fan of chemicals, I took the novel approach of manually ridding the leaves of these things by putting tape - sticky side out - around a long letter opener and brushing the leaves with it. The black things stick to the tape quite easily and are then thrown away. I did find what appeared to be a mature moth sitting well inside the plant while doing this today. I prodded it with the letter opener and it took off, hopefully for good. Haven't found any caterpillar-like creatures yet. Hope your plant made it!

Many thanks to suze9 for the expert diagnosis.

If you spray the plants with water for a minute or so, you'll be able to see the tomato hornworms wiggling around, and then pick them off -- yuck.