So I have one 4x4 bed (for comparison) and realistically you can fit 1 tomato in there, unless you're getting super compact "mini" plants. I've tried more than one and basically one will take over and starve out the rest.
So I'd do 1 tomato in 1/4 of the bed. Peppers in another 1/3rd, and lettuce in the rest. Try to arrange it so the taller plants (tomato) don't block out the sunlight for shorter ones.
Also I'd really recommend trying to start lettuce from seed - it's cheap and a very forgiving seed and you can just sprinkle it around where other plants aren't, and it'll sprout anywhere there's enough space and sun.
Hard to make recommendations if you don't specify what you're looking for. But for me personally, for tomato I always grow Sungolds because I like to snack on them (but because they're so sweet they're not optimal for cooking). For peppers, Shishito (fantastic pan fried with salt and pepper) but that's not a bell pepper, as I've had no luck growing those. For lettuce, I tend to go for buttercrunch types but don't have a specific cultivar I fancy yet.
Compost & fertilizer:
https://www.gardeningchannel.com/compost-vs-fertilizer-explained/ I avoid outright fertilizing as much as I can (though some plants really benefit from it, so I do have a bag around), and instead have been working on amending the soil via compost and commercial amendment mixes.
I don't know your planting zone, best to look up your USDA planting zone as well as garden guides for your location to find out when frost is no longer an issue, and when temperatures are optimal for planting certain things.
For example, I seeded some lettuce today. I seeded arugula Feb 21st and it's coming in nicely.
"When needed" - there's not a magic answer. When it's hotter or drier you'll need to water more (like in the summer I sometimes have to water daily), when it's cooler you may not need to water at all. Also different plants may have differing water requirements.
See above.
I have my beds set 16" high and that keeps out the rabbits, so that's good enough for me. If you have a lot of pests or bigger ones (or the chickens free range) you'll need some type of fence around it.