Figure I can answer most of this, hopefully others can find it useful as well.
1.) J2 DS-2019 is sent to TPL and TPL can offer to mail it to your address. Don't remember how long it takes to issue that DS-2019's but I don't recall it being a last minute thing if you have your ducks in a row and have all your paperwork submitted in a timely fashion. The bottle neck for obtaining your visa will be the fact that you are ineligible for a J1 visa until 1 month before your program start date so no matter how fast you get a DS-2019 you can't actually get your visa until June 1st anyways.
2.) I've looked into this extensively. Theoretically yes it's "legally" possible, but in reality highly difficult to impossible. You must have a high credit score to lease a new vehicle. If you have low or no credit there is no way any dealership will lease you a new vehicle. You can offer to pay the entire lease in cash up front, doesn't matter, you will not be approved. Also in order to be considered for approval of a car lease while on a visa your visa must be valid for the entire duration of the lease. J1 visas expire yearly and must be renewed each year, making you ineligible for a lease any longer than 1 year (which is pretty much all car leases). Often times obtaining financing from a used car dealership on a purchase will be a much less stringent process than trying to be approved for a new vehicle lease and may be a possibility on J1. Honestly though the most realistic option on a J1 is just purchasing a car outright, this is what I ended up doing with a used car. The only J1 holder I know driving a leased vehicle had an uncle who is a US citizen that offered to sign the car lease in his name.
3.) It's not hard to build credit, but it takes a fair amount of time. You will likely have to start off with a secured credit card (CC company will require a deposit equal to the maximum credit limit on the card to guarantee the debt.) After about 8 months of reliable payments they will offer to transition you to an unsecured credit card (refund your deposit) as well as increase your limit. If you have a significant banking history in the US demonstrating regular income you may be able to qualify immediately for an unsecured card through the bank you have a reputation with but otherwise will likely have to go with the secured card route. It will take around a year to build a respectable (720+) credit score, but it's not hard to do. Just use your credit card to make regular purchases, pay the bill on time and wait it out.
