Sorry for the long winded answer to this. But it is important to understand your situation and that no one can answer your questions without being an expert in your state.
Most insurance companies will reimburse your deductible if they collect from the other company. Find our if yours in one of them. If not you could pursue the other company. Depending on who they have that may be easy or difficult. The vehicle is going to be fixed either way. If your with a big reputable company I would personally file with my own to avoid the headaches of someone else company. (IE this is the time you are glad you have the best insurance and not the cheapest insurance). Also on your policy it will detail how much per day you get in rental car expenses. It is probably higher than what the other company will be required to pay (meaning you could get a much nicer vehicle if you use your coverage and not be stuck in an economy box for 2 weeks). Also note here if you have an OEM part endorsement on your policy you will want to use that policy for repairs. Standard insurance companies will use non oem parts when available and some will actually use recycles parts from other totaled out vehicles that have not been damaged.
Your first steps are to determine if the vehicle is drivable. Any leaking fluids, no working lights or hanging/dragging parts mean it is not drivable. In this case you should get a rental car right away and do not drive your vehicle. If it is drivable just make an appointment with your preferred shop and give them the claim number (from your or the responsible parties insurance company). In most states you can choose what shop you go to. All insurance companies will have preferred shops. By using one of them you will usually cut your time in shop down (they have a relationship so an adjuster will not need to come out and look at the vehicle to verify repairs).
In most cases the original estimate will change and the shop will need to do a supplement to the original quote. They will not know the full extent of damage until they tear down the vehicle. This is another reason to use a preferred shop because if you do not the adjuster will probably need to make a second visit to the shop to verify the further repairs. This can delay your repairs by a few days for each visit. It is also important to note that the rental car is only paid for during the time of repair. So drop your car off the day the work is starting (assuming it is drivable). Don’t drop it off on Friday afternoon with work starting on Monday or you may find yourself responsible for a few days of the rental car expense).
On the DV claim. It is rare they are ever paid out and many states do not recognize it. Technically you do not own the vehicle so the leasing company is who would collect the funds anyway.
On your rates raising if you file the claim on your policy. Most companies will not raise your rate at all for a not at fault claim (in most states, some states allow this practice). Some do (Progressive comes to mind in states that do not allow it by changing your rating factor–ie accident free discount goes away even if it was not your fault).
Moral here is there is no right or wrong answer and you need to do what is the best for your situation. Claims processes are very state and company dependent. Hopefully you have a good agent that can answer all of the questions for you so you do not have to rely on the internet for answers that may or may not apply to your situation.