To be honest, I have never heard of such a thing in more than 15 years of practice. My first instinct would to be to say she has no governmental authority to issue speeding tickets that would be enforceable. However, I would check your association rules to see if it addresses this issue. You can also seek consultation with an attorney who practices association law.
Categorically, a condo president does not have authority to give a "speeding ticket" that indicates a violation of any governmental law or ordinance unless the condo president is a sworn law enforcement officer. However, it sounds like the alleged violation was driving over the speed limit on a private street that is part of the condo. That being the case, the issue has to be addressed under the covenants applicable to the condo. The condo may have the ability to assess fines against members for violation of the adopted rules, and one of those rules may be a speed limit on the private streets. The covenants also may provide a procedure for warnings and for appeal of a claimed fine. The covenants form a contract between you and the condo association, and your rights to question a fine are going to be covered by the covenants. That includes whether the association has the right to tow your vehicle. If it remains an issue or comes to a head with a fine being issued, you are probably going to have to hire an attorney to review the covenants and advise you on your options.
This answer is for general purposes only, and it does not create an attorney-client relationship.
It depends on whether she gave you an actual Uniform Traffic Citation that requires your appearance in court. If so, unless she is a certified law enforcement officer, she cannot do that. If it is not such a citation, my colleague's answer above about the private condo laws is a very accurate answer.