Let's see... That's Flower between 5th & 6th...
You should be no worse walking in the immediate vicinity of the hotel than you would in Manhattan. While I would not want to do a lot of late night strolling, morning & afternoon should be fine. There's some interesting shopping opportunities in the surrounding neighorhoods (LA's jewlery & garment districts are nearby, so's the wholesale flower market if you feel the need to throw bouquets)
Incidentally, you are within a block or so of the 7th Street Metro Center, which is a stop on the Red/Purple Line subway, and the Blue Line light rail... A $5 daily metro pass allows you unlimited use of both (plus METRO busses and DASH shuttles) ...or you can pay $1.25 per boarding. There are LOTS of dining possibilities along the Metro lines in question:
See the Metro Dining guide.
Additionally, there's a lot of good ethnic possibilities. Little Tokyo is just a DASH ride away. Koreatown is just down the Red/Purple line. Lunch at Grand Central Market elevates Mexican fast-food to a whole new level.
Incidentally, the Staples Center is just one stop down the Metro Blue Line from your hotel. You could easily take the Metro back when it gets out.
Not a lot of places I'd want to run in downtown... But if you have a car, Griffith Park is just 5-10 minutes north of downtown, with miles and miles of open space, jogging and hiking trails, golf and tennis facilities, and even riding stables if you feel so inclined.