1. Will we then have two bank accounts (our regular account and a separate online account)?Only if you use Paypal: verisign, iBill, icverify, etc link to an existing checking account since they don't run their own banks (or whatever Paypal is this week, a mutual fund?)
2. Do I need to buy an SSL certificate or would the company like verisign be an alternative to that?Nope. The third party processor negotiates directly with the customer, unless you need to integrate a homemade shopping cart or something (you don't).
3. If all we are doing is taking donations, then I don't need any special shopping cart software, do I?Correct. You'll add a link or a <FORM> at worst, unless you anticipate being amazon.com.
4. Is Verisign the company I should go with? No. Verisign's the most expensive, you pay hand over fist for brand name and squishy secure feelings; they're as bad as NetworkSolutions, so bad they had to acquire them to retain the title of "worst tech support to cost ratio in the industry".
http://www.ibill.com charges a buck per credit card transaction. Paypal's considerably cheaper for small transactions (3% plus .30). So a $10 donation would be $9.40 after Paypal, $9.00 after ibIll, and considerably less with Verisign unless you do a thousand like it every month.