if you're "across from the standard" that suggests that you are staying in someone's apartment at the pegasus. hopefully, they're loaning you their parking spot too. but even if not, downtown LA is very well integrated into pub transit, with the dash as noted, and the red/purple/blue lines are a block away.

but your questions were (1) restaurants, (2) safety, and (3) workout areas.

there are lots of restaurant options, really too many to easily list. what are your preferences?

downtown LA is pretty safe. there are always people about (i've been out as late as 2am and as early as 4am, and never felt unsafe).

for working out, the pegasus has a rooftop pool, but i don't know about any fitness center. i've seen people running downtown, but i've always thought they were crazy (i personally hate running where there are so many cars and buses). perhaps you could look into the LA athletic club (on 7th and grand, i think).