Having a dog is going to limit you – not only with apartment or townhouse living, but because it is so competitive out there for rentals, most specify that their property isn’t pet friendly. You will be able to get a unit (1 bed, 1 bath) in the central suburbs for the top end of your budget but obviously this isn’t good for pets. I have had a look at some of the more central suburbs for you and there really isn’t much.
Will you have a car? If so, then the whole central thing won’t matter as Canberra is very easy to drive around once you get to know the city.
Now onto the other things
Cyclocross - We have a series here that is put on by the Canberra Off Road Cyclists (CORC). http://corc.asn.au/tag/cyclocross/ It gets a good following and is run by some very passionate CXers
Bikepacking - Is alive and well with many groups and individuals heading out for a ride. BigAdz is your man to talk to here.
General booze filled shenanigans - Canberra has a good selection of unique bars but one of the things that we don’t have a great deal of is ‘the local pub’. We do have sports clubs with pokie machines (blergh) and bizarre taverns that are a bit on the rough side. I am a little unsure of what else to say here, it really depends on what you are after. We have access to great craft beers, a few breweries in town and in Summer there is a weekly meet up of a bunch of cyclists for a bit of a ride and a couple of froths after. Like most places, Canberra has it’s own personality that you will grow to love after you understand it.
I am a huge fan of the city, I have lived here my whole life and feel no reason to leave. It has everything I want and as you would know, the beach and mountains are close by.
Have a look through that allhomes website. That is where you will find all available rentals. Shout out if you have any questions for particular suburbs.
Oh, I forgot. There is a Canberra division of Engineers Australia - not sure if they would be helpful for you to ask a few questions for employment in the ACT. I recently completed some signage work for their building, so you know it looks good.