Can a green screen key done in a browser look professional?
Yes; the result depends on the footage and the matte work, not on the machine the keyer runs on. The browser pages include the qualifier, edge tools and despill a working key needs.
Why is there a green rim on the edges of my key?
That rim is spill: green light bouncing onto the subject. Despill tools subtract it, and shrinking the matte slightly also helps. A rim light at the shoot prevents it in the first place.
Do I need a big green screen at home?
For waist-up shots, two metres wide by two high is workable. Full-length shots need the screen to reach the floor and beyond the frame; folds and shadows are what give small screens away.
What should a composite be exported at?
The same delivery preset as the rest of the project, unless the composite feeds further effects work, in which case a master grade file like ProRes keeps room in the highlights.