None of my kids had stage 1 for more than a day. My 2nd and 3rd child went pretty much straight to stage 2 consistency. It's easier for them to handle becuase the stage 1 is just too runny for them to move around their mouth with their tongue. They all did much better on chunkier food. That said... my older 2 were pretty much done with stage 2 consistency food within a month of starting solids. They were doing finger foods (we just cut everything we ate into pea-size bits) or fork-smashed food by 6 months old (start cereal/solids at 5 months). The older 2 were 100% self-feeding (no spoons, everything can be a 'finger food' though) by 8 months. And no, they don't need teeth -- none of my kids had a tooth before their 1st birthday.
My baby is 5 months old and we're doing pretty much the same thing food-wise with her... she's eating oatmeal (real stuff from the canister not nasty tasting rice stuff), a variety of fruits, veggies, meats and pasta. Just yesterday I gave her a pea to see what she would do with it... she picked it up and popped it into her mouth and chomped away on it with no problems at all.
There's no need to wait a week between new foods. Just cut up or fork smash whatever the rest of the family is having. Many studies have shown no corrolation between allergies and when foods are introduced. Other studies show that it often takes many exposures to a food to develop an allergy -- alot more often than they'd get in a week. So even if you wait a week you could still be spending alot of time figuring it out because it could be something introduced weeks or even months earlier. Less than 6% of people have a food allergy.