if we've set a bid of $13 for a particular keyword while actual keyword bid for that day is $9. ... So, my question is would Google charged for $9 or $13?
Google will often charge you less than your actual bid, sometimes a lot less. If yours is the highest ranking ad, you'll only be charged enough to beat the second guy, not necessarily the full amount you bid.
The rank order of ads is based on a combination of the amount bid and the quality score, which depends heavily on the clickthrough rate. You can boost your ranking by bidding more, OR by writing a better ad and/or improving your targeting so that you're getting a better clickthrough rate than the other guy. It's well worth while to test some ad variations because if one of them gets a better user response you can get more traffic for your ad spend.
Think hard about whether you want to be at the very top, where you'll get curiosity clicks from people who just click the first thing they see. Sometimes the conversion rate and return on ad spend are better if you ride lower in the ad column. That will be something to test.
Something else to think about is searches where you don't want your ad to appear. Sometimes a search can include words that you're targeting but nonetheless be off-target for you.
Example: if you were selling blue widgets you'd normally want to bid on searches about blue widgets, but what if the search were something like "blue widget product recall" or "blue widget lawsuit"? Those users are likely poor prospects for buying your widgets. You could block such searches by adding the words "recall" and "lawsuit" to your negative keyword lists. Spend some time with a keyword suggestion tool, looking for searches to bid on but also keeping an eye out for searches that would be off-target. Build your negative keyword lists.
I always, always block the word "lyrics" for search campaigns. ;)