First off, ask your trainer how many athletes he or she has trained that have won national, regional, or international titles. Personally, my belief is that unless you have trained someone that's made the Olympic team (or around there), you should probably put your ego in check and emulate what the Olympic trainers are doing.
Training isn't an art project where there's a lot of room for individual expression. Different athletes respond in different ways, but most things are quite concrete and should only be changed when appropriate peer reviewed research indicates as such.
Further, if you look around a modern gym you'll see a lot of trainers doing ridiculous things with their clients. In my own observational experience, trainers are basically motivators and task masters, not people I would trust with exercise science before I observed them first. This the trainer's fault of course, but it's not made any easier by clients who don't really care about pushing boundaries and primarily just want the fore-mentioned "someone to tell me what to do in a gym because I hate going."
Static stretching before exercise is not considered helpful:
The basic science literature supports the epidemiologic evidence that
stretching before exercise does not reduce the risk of injury.
Additionally I'd reference Ivo Flipse's excellent answer from 2005. Stretching for performance is still being figured out, but there is certainly no credible evidence that static stretching before weight training is a good idea.
This one I actually think your trainer has right, or somewhat right. It is important to warmup, both your general body systems and the specific body components that you intend to lift with. There is still some squabbling over the benefits of stretching, but warmups are universally acknowledged to be critical to injury prevention, force production, and mental readiness.
20 minutes might be on the high side, but as a 40-ish year old man, I've adjusted my warmups from 3 minutes to 10, and remain relatively injury free. I actually need to "warmup" into my "warmups", if that makes sense, hitting the jump rope for a bit before I do getups before I do air squats, before I grab an empty barbell.
Dealing With Your Trainer
Your trainer is probably fine for a weekend warrior with no real physical ambitions, but I think you can quickly realize he or she isn't going to place anyone on an Olympic team any time soon. You'll also notice that you probably don't have a lot of high level athletes in your gym. These people train and work in difference circles.
Getting a good trainer is going to cost you, you'll need to switch gyms, and you'll need to travel. If you don't want any of that it's okay, but realize that much better training is out there: most people don't pursue it because it's just not enough of a priority.
Trainers who place two national level athletes qualify for USA Weightlifting's National Coach standard. You can search around for USA Weightlifting clubs by going here. Many of these clubs have turned into Crossfit boxes, which is a whole different topic. But if you live near a major metro area, with a phone and a day's time, you should be able to talk to someone who (unlike your trainer) is truly qualified to get you to the next level of fitness.
Just be blunt: ask them if they are USA Weightlifting licensed coaches, and ask if they are (or know any) National Coaches. I just googled around my own hometown of San Diego, and a resume like this is what you're looking for. Concrete results, proven history, and official licenses from strong organizations.