You have a much better chance of getting a slot if you enter a lot of dates on your application. Here's how it works:

In a random process, your application is pulled up to be processed. If yours is one of the earliest pulled up in the process, you have a very likely chance of getting your first choice day. If yours is processed later in the run, after most (or all) dates are filled to the quota, you are out of luck.

When your application is processed, if you have specified only one date, if that day is full, you are out of luck. If you have specified other dates, the process checks each one, in the order you entered them down to the last one, checking for an open date. If there is an opening on one of the days, you get that slot.

Here are Lottery results from 2013 go give you an idea of your odds:

year 2013 2015
total applications 12,153 11,662
successful 4,591 5,015
unsuccessful 7,562 6,647

When specifying dates, understand that weekend slots fill up first, so if you enter lots of Mondays through Thursdays, you have a better chance.

Last year, about half the lottery entrants actually received a slot from ALL their choices.

With your lottery entry, I am pretty sure you can go back and revise it, before March 15, and change or add more dates to it at no extra cost.

As for parking, Whitney Portal is the only area affected. If you want to acclimate, you can go up to Horseshoe Meadows or Onion Valley for a day or two before hiking.