On the " how likely am I to get approved" question, that depends on your debt to income ratio, but you will drastically improve your chances with that proposed 20k down payment. Just about anyone can get financed for anything, with enough down payment.
High 600s is not a bad credit score. Usually, lenders offer their "best / lowest" rates for people over 720, but high 600s is usually fine for getting approved for a loan in general... keeping in mind things like debt to income ratio, etc.
As far as "hits to your credit" for applications, certain types of credit applications are lumped together as 1 "hit" to your credit score, because its acknowledged and expected that you will apply multiple times for that loan, but only take one. All applications over a specific time period show up on your credit report, but the reduction that happens for applying for credit only happens once. Home loans and car loans are the two that come to mind that work this way, and I believe the period is 2 weeks.
Meaning, you can apply at 6 places for a car loan and as long as you do so over a compressed 2 week period, the hit to your credit score will only be once for those 6 applications. Credit cards dont work this way, they hit your credit for each one applied.
Since, as was mentioned, many credit approvals only last for X amount of time, its best to apply "closer" to the pickup of your car. I would recommend that you identify the financial institutions you want to apply for a loan with (your bank, maybe a credit union, maybe tesla), and just keep your "ducks in a row". Dont apply for any credit anywhere while you are prepping to get your car. Dont apply for any new credit cards, any furniture store credit, nothing. Keep your balances on your credit card(s) stable, and dont pay the minimum payment on them (this is good general advice but more important if you are trying to make yourself look "better" from a credit perspective).
If you dont have your down payment now, and put down your deposit now, you run the risk of them possibly matching you to a car and you having to say you are not ready. You could go to a tesla location and order with a person, they may be able to note your account that you are looking for pickup around XXX date, I dont know.
You will have time to execute (apply for loans) when they tell you the car is ready. You wont have to come up with any additional money until you go to pick up the car.
Just keep your credit in line, get your down payment together, pick your preferred financial institutions to apply at, and apply close to when you are ready to get the car at all of the places you identified you would accept a loan from. Applying for a car loan takes almost no time, almost all places do it online now, and you get decisions either instantly online or within 24 hours. You will have time, and approval will likely be no problem, with a down payment that large.
***I am not in the credit industry at all, I just learned a lot about it previously when going through rebuilding my own credit 10-15 years ago.