Automated backups are a significant drain on the system. If it can automated, everyone will just enable it. I'd prefer if you guys could try and remember to do it yourself.
Why not try a free "co.cc" domain? They're a bit more professional. The link to signup with them is available on the HelioHost website. Once you have one, you can park it from your cPanel.
Looks like you were suspended for either overusing crons or for overusing system resources. Do you remember any crons or resource-intensive scripts you were running?
No - consult Google Apps for help setting it up on a cPanel platform. They should have a help page, considering that the plurality of hosting services utilize cPanel.
Create a new FTP login if you'd like. Making things easier for you is not my job. The sign-up clearly stated you only had one chance to create your account.
Ah - Google is actually having you set the same MX record, but just under multiple servers with different priorities for failover. Go ahead and just set the first record, and you should be fine.