So Biggs and a Firespray are touching after activation is over. One cannot target the other. So does this mean the Firespray is free to attack my other ships at range 1 away from biggs without worrying about killing him first?

Yes. Since you can't target Biggs, due to touching bases, you are allowed to shoot any other target you have in your fire arc(s)

Yes. You must target Biggs if able unless he is more than range 1 away from your target or you cannot pick him as a legal target.