Possibly. But the odds are dramatically reduced compared to no condom. Warts can appear at the base of the penis where the condom did not cover for example. Urethral infection with organisms such as gonorrhea and chlamydia are less likely to be contracted unless the integrity of the condom was compromised. Abstinence is the only certain way to avoid std, but condom use is the next best thing.
Yes. Yes, far less likely than without, but yes. Condoms are not 100% effective and there still is a small risk you could be infected even with condom useage.
Unlikely. Probably not. Using condoms is actually one the best methods to prevent sexually transmitted infections.