How grout entire floor without disturbing the grout?
You need to start at the furthest corner from your exit path and work your way out. Grout the entire floor at the same time or you may see color differences where you stopped then started again. Mix all the grout as a single batch (that is a fairly small area, it should not be too difficult).

Use your grout float to remove as much excess as possible as you work. I keep several buckets of water and several sponges and do a quick damp "dirty" sponge to the grouted areas as I work. You can use a spray bottle to give a slight misting to completed areas if you are concerned that you cannot work quickly enough. Cover unused grout in the bucket with plastic wrap, you can use this to repair any grout lines accidentally disturbed during the next steps.

When complete, let it set for 15/30 minutes then proceed with your sponging. It is important that you keep a "dirty" sponge and bucket (initial wipe), and a "clean" sponge and bucket ("clean" wipe). Wring out and rinse sponges after each wipe, and change water frequently.

Let it set overnight, then do a "haze sponging". Mix a small amount of vinegar (or muriatic acid) with clean water in a bucket and sponge the entire floor to remove remaining grout "haze".