1. Yes, Burpee's common basil's seeds can be saved and used. They will stay viable longer if they are cleaned to remove all the chaff (I'm still using sweet basil seeds collected 6 years ago).
2. IMO depends on how you grow them. If you pinch plants to make them bushy, more space is needed. If you just let them grow with one stem they can be very close (1-2'').

If your seeds are for a hybrid variety, you may find that the offspring from seeds saved doesn't come true to the qualities of the parent plant. It is just how hybrids work. But you can always try the seeds and see what you get.

The next time you buy basil seed, look for heirloom or open pollinated varieties for best seed-saving results. The seeds saved from these types of varieties do produce the same quality and characteristics as the original plant. While they may not be as common as open-pollinated or heirloom vegetable seed, there are OP and heirloom basil varieties available. I like "Siam Queen" which is an OP/heirloom Thai Basil type very much. There are other varieties and types to be sure.

FataMorgana