Gastown
It should be walkable from where you are staying even with your bad foot. The area in between is not the greatest but it's not overtly bad. Just hopping on the Skytrain might be simplest.

A stretch of Hastings Street (~btw Main to Cambie) and a bit of surrounding neighbourhood is notorious for being one of the poorest areas in Canada and drug activity. I wouldn't necessarily call it unsafe from a violence perspective (I'd walk it if I'd have to) but it's not a pleasant area and why take the chance.

Chinatown
Like gglave said, probably easiest just to walk. However, where you are staying is a bit higher in elevation than Chinatown. You'd have to slope down along Beatty or take the stairs around the stadium.
We park in Chinatown and walk to the stadium for games and find it a light walk.
Chinatown is not as vibrant as it used to be. In the past, Chinatown used to be pretty much to only pace to get Chinese goods/groceries and attracted the crowds. In over the last decade, Chinese shopping has been more distributed with a large hub in Richmond (without the historical/cultural factor).

Granville Island
Like gglave says, yes, you can walk across the bridges but you have to watch for traffic when you cross the on/off ramps. Burrard St Bridge is probably a more scenic route if you are going to go to Vanier Park also.
Personally, if it's a nice day, I'd recommend just getting a daypass for the ferry that runs around False Creek.

Vancouver Zoo, UBC, QE Park
- Never been to the zoo myself. It's definitely a bit of a drive. Not worth the effort IMO.
- Yes, you can get to UBC by bus. Easiest is across Broadway with a transfer at Alma I think. UBC is a pretty spread out campus. It's a bit of a (triangular) walk btw the SUB (Student Union Building), Museum of Anthropology, and Botanical Gardens. BTW, UBC has a Rose Garden too.
There's campus parking lots all around (Botanical Garden, Museum of Anthropology has metered but the Rose Garden across the street has underground parking, etc)
You have to like West Coast aboriginal art to really enjoy the Museum of Anthropology.
The Botanical Garden is very... woodsy and plantsy. Not very flowery if that's what you are looking for.
- QE park would be a more compact attraction. More garden/flowery.

Burnably Village Museum, VanDusen Botanical Garden, Police Centennial Museum
- Personally, I don't see the attraction of Burnaby Village Museum. We've had a couple of company picnics there and I found it to be ...meh. They have a carousel. Nearby Deer Lake is nice. Hart House Restaurant is nice. No worth the effort to get to without a car IMO.
- VanDusen is easier to get to by bus. Maybe latch it onto a trip to QE Park?
- Police Centennial Museum... never been.

Not familiar with Smart Visit card (tho I took a quick look at it online just now and it looks kind of iffy from a value perspective if you ask me). Haven't taken the trolley but seen it around town. Not familiar with the Big Bus.

QE Park is along Cambie St where they've been doing a lot construction for the new airport to downtown skytrain line. Haven't been to that neck of the city for a while but I think while there still be some construction going on, it's not as bad as it was.

Agree with gglave that the area around your hotel is very safe. You're pretty good heading every direction. But once you head N (ie. NE) across Dunsmuir, it gets a bit seedy IMO but not necessarily unsafe.
If you want to head out of the hotel for the night and check out the surroundings, your best bet IMO is to head W (ie. NW) along Robson and then branch off to what interests you.

Zachary's, Dining, etc
- Maybe use Dinehere.ca as an initial guide. Take some of the reviews with a grain of salt.
If you like Belgian food (ie. Moules Frites, etc), I'd like to recommend Chambar, ~4 blocks from your hotel on Beatty.
There's also a 7/11 down Beatty on the corner of Beatty and Smithe.

Heads up that there's a pre-season football game at the stadium on Tue the 23rd.