Hi CK123,
Here are my thoughts on some of your Q's:
Overall, I think you've picked some great locations to consider - Avondale, Ortega, San Marco/San Jose, Southside, Intracoastal West/Beaches - all are nice places to live.
Commute-wise, since you'll be working downtown, obviously the river neighborhoods will give you your best commute. When I worked downtown, I was living at the most southern end of the San Marco area and my commute was 12 minutes (I was heading in to be at work around 7:30/8:00am).
This is not to rule out Intracoastal West/Beaches. Given your budget, it is unlikely you'd end up at the beach itself, and that is where the really long commute comes in. So, if you just consider the area west of the Intracoastal, your commute would probably by 30 minutes+ depending on how far east or west you go in Intracoastal West.
When considering your budget in regards to a single-family home, if you end up looking at homes for $175k or less, I would suggest the San Jose area (32217). $175k will get you into a nicer neighborhood in San Jose than it would in Intracoastal West. In IW, to be in the nicer neighborhoods, you want to be in the new/newer sections and the entry cost is higher (more like $250k). In San Jose, the old housing stock is nice and even a modest $175k house will be in a good neighborhood. If you want to be on the Westside, over in the Avondale/Ortega area, try searching in Fishweir Creek area - there are some nice, modestly-priced homes there (sorry, I don't know the zip).
As for condos and townhomes in Jax, they can't give them away now - the market is so flooded. Personally, I think your money would be better invested in a single-family home. I do know some of the ones you mentioned though:
Drayton Park is nice. In your price range, you'd probably end up with one of the townhomes in the back (they look a lot different than the pretty ones up front by the road

). The townhomes in the back don't have garages and are a lot smaller. The ones up in front are gorgeous and huge (one of my friends owns one), but they cost much more $. The location is very convenient and there are some great restaurants right next door in the Publix shopping center (Crazy Sushi, Sangria House...). Touchton Road floods and it can be hard to make a left out of Drayton Park when it's busy (no traffic light for your turn). People do have dogs there, but there's no real place for them to play/run around (they're not allowed to play on the greenspaces, from what I understand

).
The Colony at San Jose is in a spot that is not so great. It's a good part of town overall, but that particular spot is not one where I'd want to live (sort of the worst location inside of an otherwise good location....if that makes sense

!).
Pottsburg Crossing is in a good location - on Bowden Road in the SouthPoint area (lots of people work in SouthPoint). It's easy to get onto I95 from Pottsburg Crossing, Phillips Hgwy/US1, even Beach Blvd. So, it's a good central location. The surrounding residential is okay, but the zoning is a bit funky. It would be a quiet spot at night, but busy during the day. Making a left (which is probably almost always the way you would go out of Pottsburg) will be hard when it's busy - you'll have to see what I mean when you see the way the entranceway is designed. I drive by here a lot and the entranceway is less than ideal (people going in and out of Pottsburg get in my way a lot

).
With a dog, I think it's nice to have your own backyard. You'll find some nice homes in San Jose (32217) in your range, they just might need some updating/work. You can even try searching a bit north into 32207 - you might find something in Lakewood or Ridgewood, but for $175k, the house might be really small (under 1000 sq ft) - the further north you go, the more $ per square foot in this part of town.
Oh, and Arlington is not bad. There are some nice neighborhoods in Arlington and your money will buy you more square footage there too. Arlington is a big area, and there definitely are some bad parts that make the news regularly, but that certainly isn't the whole story of Arlington.
Hope that helps

.