Pokemon-go – give the professor the Pokemon in order to evolve

Yes, they do. Once you catch one, look at their record in your Pokemon list and you'll see a button in the middle labeled "Evolve". Next to it is the number of the specific candies needed to evolve them.

You get the candies 2 ways:

1. By catching more Pokemon of that same family (such as Rattata and Raticate), this gives 3 candies
2. By "Transfer"'ing duplicate pokemon to Professor Willow (By clicking the "Transfer" button at the far bottom of the same screen). This gives 1 candy

I believe it does. What changes when you progress up the Levels is the Max CP a pokemon can have. If you're feeling sentimental towards your starter you can upgrade him over and over again, with candy and stardust and will eventually be as strong as a wild of the same starter type you find at Level 30 (current Wild max CP cap).

But if you don't hold any special feelings towards a said Pokemon always transfer the lower CP Pokemon and keep just 1 or 2 with the highest to evolve later on.