Will desensitizing me to penicillin reduce my penicillin allergy?

Yes, but... Desensitization is not permanent. You have to be desensitized each time you need to take penicillin, if there are no effective alternative antibiotics that could be used. Your state of desensitization will continue for as long as you take penicillin every day, but you'll have to be desensitized again if you don't take penicillin for more than 48 hours in a row.
Yes. The way desensitization works is that you are given very small doses of penicillin initially. The amount is gradually increased to get your body not to react to it. There are very few infections that require treatment with pcn so desensitization is rarely done.