I don't think you can or even will need to. The insurance companies have a pretty good idea what a realistic price is going to be to repair your car. If you really disagreed with what they come up with, you can get your own estimates and try to argue that way, but I don't know how far that would take you, since they likely have shops they partner with in your area and the adjuster is an expect at determining damages; its what they do for a living. If you really don't like their number, you can always submit to your insurance company. Then you'll be dealing with them, and they'll deal with the other insurer.
Ask for a rental; that's a perfectly reasonable request and I'd be surprised if they didn't cover one. It should be comparable to your car, although I wouldn't try and get a BMW or something. Think same class (sedan, SUV).
I also would be surprised if your insurer raised your rates for something which was not your fault. Its possible I suppose, but your insurance probably requires you to report all accidents anyway, and they might have a clause saying they can drop you or deny coverage if you fail to. You may have accident forgiveness too, its a pretty popular thing to be just included now, unless you have a really barebones insurer. Your best bet is to call them and let them know what's going on, and that you're dealing with the other insurance yourself (for now).
I had to take a half day vacation from my work (lost wages). I feel
disoriented and I am not able to focus on anything(after accident
shock) even though I feel fine physically.
I think no one is going to care about this stuff. You're not injured, you didn't lose wages (unless your in CA, vacation days aren't considered wages), and it doesn't sound like the accident was that bad (I don't get the impression you believe your car was totaled). Accidents are part of owning a car. You need to deal with it and move on; there are much worse things in life. Be glad you're not hurt.