Although I try my best to learn fast and not to annoy people with my
dumb questions, I can't avoid making many silly mistakes.
Should I tell my manager that I ought to receive a proper training
instead of just learning along the way?
Should I tell my colleagues to be honest about the feedback they give
me? My concern here is that I'm getting light treatment just because
I'm a woman.
Hmm. You've only been there 3 months. It's possible that the situation is exactly as you perceive it, but it's also possible that you are reading too much into a short-duration situation. Try not to jump the gun here.
Everyone can avoid making silly mistakes. We just need to be more careful and learn from our past mistakes. And what we perceive as a big deal, might just be normal for someone who has only been there 3 months. A 2-year age difference is nothing in a corporate setting. Don't overreact.
Try to find a mentor. Perhaps one of those folks who are 2 years older than you was right where you are now just a few years back. Find such a person who might be willing to talk with you periodically. Ask for feedback on what you are seeing, feeling, and learning.
Telling your manager that you ought to receive proper training is a bit presumptuous. Such a declaration ("I expect this... and what I've received so far isn't 'proper'...") might not be well received.
Instead, talk with your manager during a one-on-one meeting. Explain how you feel you need more training to be your best. Ask her how best that could be accomplished. Your idea of "proper" and her idea may not be the same. There are many ways of being trained - self-learning, mentoring, inhouse training programs, external training programs, etc.
Are you sure you are getting "light treatment"? How do you know? And are you sure it's "just because [you're] a woman"? How do you know? Try to bounce those thoughts off of your mentor, or someone else you trust who might provide some insight on the situation. You don't want to react to a situation you have misjudged.
Overall, try to be patient. Three months still puts you in the "total newbie" category in many shops. It might not have anything at all to do with your gender or your age - just your newness - and newness wears off over time all by itself.