Answer: First, don't use weed killer if you plan to plant in the area. It's easier just to remove the lawn by digging under the roots. Take a garden spade or shovel and cut down into the grass and straight down through the roots. Bring the shovel up and set it down right next to the line you just created. Do this for 2-3' in a straight line. Then start making another cutting line through the turf about 12" away and opposite of the first line. Your goal is to make a 12" wide strip, 2-3' long. Once you've done that you should be able to manuever your shovel under the edge of the lawn, beneath the roots, and roll the lawn up from the area you cut. It will look a little like a slab of sod like you see in garden centers. Do this as many times as necessary to remove the grass from the area in which you'd like to plant.
Neither peppers or tomatoes will grow well indoors so your best bet is to plant varieties with short maturity times. Patio tomatoes, for instance take only 60-65 days to mature.
To answer your third question, turfgrasses grow fast, regardless of type. You might want to growing groundcovers instead of grass. There are many, and they require a lot less maintenance than turfgrasses.
Best wishes with your landscape!
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