I have a 100 AMP breaker box that is beyond maxed out. I have a lot of the breakers doubled up and I have 1 piggy back 15 and 1 piggy back 20. Is it possible for me to upgrade my box without getting new wiring from outside?

I am remodeling a bathroom and purchased a new exhaust fan. I Thought I could handle this simple replacement, but now have questions. First, my house wires have no ground. Do I simply not use the ground provided with the fan? (One end is grounded to the metal housing, the other end would be for me to use) Or should I run a new wire and ground it somehow? Also, the instructions tell you to use wire of the appropriate rating? My house wiring is 14, but I have a book that recommends a bathroom exhaust fan use 12 AWG. What is appropriate? I originally planned on using the old 14 already present for the old fan, assuming it is not wise to connect 12 AWG to the old 14 and making my job a little easier. However, now that I have removed the old walls I see that is exactly what a previous electrician did. At some point in time they hooked up 2 12 AWG wires to the old 14, one was for a light fixture above the medicine cabinet and one to a GFCI. Their actual connections are hidden by a bunch of electrical tape (don't worry, there are wire nuts under the electrical tape). I am wondering if is it safe to use the 12 connected to the 14 and how the GFCI is really grounded since the wires end in this little glob of wire nuts and tape? The GFCI was working before I removed it, but if it is unsafe it would only make sense to fix it now while the walls are torn out. Thanks for your help.