You would set your ads to show more than one ad in the manage settings >> ad and put 3 for how many per location but this is sitewide. I believe there is a feature request to change this.
The other thing you could try is putting the ad codes in one block rather than in as ads.
Do you mean the existing template's search box, or Google search?
I previously succeeded in getting a Google search box to work in block 3, but couldn't get it to work in block 7. Plus, it's messy unless you pay to have ads removed.
The nice, clean search box I have at the very top of the template, by the logo, would be great if it were visible even when not logged in.
This is one of those huge DUH things for me; can't believe I never noticed it before and it was brought to my attention by a few members.