1) Are there alternatives to this - is there a 3rd party provider who will give me this service without me sending them the credit card info?
Only Google offers Adwords advertisements on their website. So if you want to have your ads appear on Google's website you must do it through Adwords.
2) How secure is it to send my credit card fax via some online fax service?
It's secure as long as you are sure no one is copying the fax. This really is only an issue on your end so if you can watch the fax process and ensure no one makes unauthorized copies of your card you should be fine. Doing this is not uncommon.
3) Do you think they will reject the application if I hide my CVV number in the fax?
Probably. There's no reason to hide that number so if you do it probably will raise a big red flag with them. There's a lot of fraud involved in advertising so anything that makes you look suspicious is probably going to hurt your chances of being approved.