Any sprinkler system that soaks the soil will work. The advantage of drip is water conservation and less chance of spreading disease. Keep in mind that veggies only need about an Inch a week (dependent on weather conditions) so you shouldn't have to water every day. Its better to water deep less often than shallow every day.

Water hitting the leaves can spread disease problems around to other plants, but Ma Nature does it, so there is nothing wrong with it.

Use any good garden fertilizer. 10-10-10 is easy to find and pretty cheap. Any thing labled for veggies would work. Keep in mind that peppers DON'T like much fertilizer, but a shot of Epsom salts can help them get some essential nutrients.

Do not water the leaves. Your veggies will be much happier if their leaves are not regularly wetted. A drip system would be nice but with a garden that small you might just as well use a hose-end soaker of some sort and do it by hand.
It's hard to say what's in the compost and topsoil you added. I would wait and see how well things are growing before adding any fertilizer. In fact, that's a very tough question indeed and impossible for anyone to know without detailed knowledge of your soil and amendments. If anything, I would sparingly add some organic fertilizers to avoid overdoing it.

Drip is always better than overhead watering since wet leaves promote powdery mildew. Cukes and squash are especially prone to it. But if overhead's all you can do, it's better than nothing for sure, and definitely best done in the morning so they leaves can dry off quicker than if watered at night.

I agree with Tcstoehr about waiting to see how well things do before adding fertilizer. Too much nitrogen can make your plants have tons of beautiful green leaves and few fruit. Veggies are just like us ~ feed them too much and they get fat and unproductive. ;) I like to use composted manure, an inch or two on top of the soil as a mulch, as my main fertilizer. It's the original slow release fert that contains just about everything your plants need. I do sometimes have to add something else, usually fish emulsion or seaweed, but I only add those after the plants have "told" me they need it. Google up images of nutrient deficiencies (all the big three as well as "micronutrient deficiency" and "mineral deficiency") so you'll recognize it when you see it.

Good luck! :)