https://www.amazon.com/gp/product/B009GU4RMC .
Fills all of your requirements. Don't bother moving the plants around, just plop the seeds under this for 16 hours a day (I have mine on a timer).
FYI, you may be behind on your timing: I started my peppers February 1 and I won't be planting them out until May. Tomatoes are a little faster than that, and eggplants are even slower.
Not really easily, but the shelves could be adjusted. They go up and down in increments of one inch, but it's a pain to do. Instead my plan is to have shelves of each different height I need, and move the plants around based on their current height. Once a plant is tall enough to break into the next compartment, I figure it's got plenty of light anyway (see my tomatoes - light along the way).
Yup, pretty good coverage, at least when you have a bunch of them and the area is surrounded in reflective mylar.
Looks like you didn't see both of my T5 links. I have both these hanging 2' T5s and also these freestanding 4' T5s. Both are adjustable and offer good coverage, about 1' each way from center.
Remember LEDs produce more light per watt, so for apples to apples, you need to look at the lumens, which unfortunately not all lights tell you.
The fluorescent tubes are definitely brighter per-foot than the LEDs.
Yup, can't hurt to try * , and if you decide you don't like them, they might make good under-cabinet lighting.
* This is not medical advice. If this were medical advice, it would have been followed by a bill.
It depends on the plants, but that light wouldn't be much help to most garden/outdoor plants. Even something like this would need to be right next to the leaves to give them enough light, and left on for at least 12-16 hours.
If it's an indoor plant though, they generally don't need as much light and will most likely be fine even with that bulb you linked.
I started around the same time... last spring/summer or so. All Kratky so far, but looking to experiment with some DWC soon.
I used 2 of these hanging T5s and 2 of these freestanding T5s for nearly a year with great results. Strongly advise using such an adjustable solution as you can keep them the recommended ~6" above the leaf canopies.
During a recent upgrade, I just switched to 42 of these LED T8s (well, 7 of these 6-packs) to reduce the amount of space being used by my lighting solution. Good so far after about a week; looking forward to the longer-term results. One six-pack of these covers about the same amount of space as one of the the 4' T5 I linked above, and takes up much less space. Downside: No reflectors. But I enclose my grow areas in mylar, so meh.
Whatever lights you buy, make sure they're the right color temperature for your kinds of plants. If you're growing many different kinds of plants, make sure they're full spectrum.
Additional advice: Once you get some mature tomato plants, transplant cuttings from their healthiest-looking tips. Those transplants seem to produce more quickly, and more total tomatoes, because they're already mature when you plant them. Took mine about 3-5 days to put down their first roots. Remember to cut at a 45 degree angle. I don't know why, but that's what everyone says to do.