Had a look at the Yarra Trails thread - seems like there is a thriving community that I'd love to be a part of. Fingers crossed that we can find something in the Heidelberg area (closer to city appears too expensive).
How is the city traffic in the morning between Yarra river trails south exit and St. Kilda from an MTB point of view? I always try to avoid inner city roads with a lot of traffic when on my bike. I've just had too many near death experiences and prefer to be safe. Not sure about VIC, but QLD car drivers will go out of their way to cut you off and or run you off the road. They seem nicer to MTBs than Road bikes, but still.
Yeah, the YT thread & the trail community is well & truly alive & kicking.
Plenty of potential for you guys in that Heidleberg, Bulleen, Thornbury area (as too Fairfield/Alphington/Balwyn-ish area) & they're all 100% manageable as far as the bike commute goes. For 4yrs I did the daily bike commute from Ivanhoe to Middle Park for work- see the couple of images attached below with the route I used to do- excuse the rough MS Paint. Give or take, you'd be looking at a 45-1hr commute time each way. There's still a few spots (mainly Heidleberg West) which are a bit sketcharoo as far as neighbourhoods go, but otherwise it's an incredibly well amenitised area (City of Darebin region) & all the infrastructure you could need as a young family in terms of clinics, childcare centres, shops & public transport options. If you can justify it, then maybe consider not mentioning you have a dog or any pets when looking for a rental as, once again from my own experience & difficulty, more often than not Melbourne real estates will simply stone wall you. Whenever our inspections came up my partner & I would simply clean & air freshen the wahzoo out of the place (usually by the scent of delicious baking etc) then throw the dog & its belongings in the car & one of us would disappear with the pooch until it was safe to come back.
One of the other cool things about that area is that you're also within coo-ee of Plenty Gorge which also has a very active thread on here & an extensive network of singletrack.
Basically everything from Bulleen (& north of Bullen within that distinctive green belt) to Yarra Bend Park/Richmond is dedicated pathway that shadows the river & has an absolute swag of ST options coming off it (heck, you could comfortably put together 30kms of ST before even getting to Richmond). Once you've crossed the Walmer St bridge at Abbotsford/Richmond you're then on dedicate pathway (cement/asphalt) that shadows the river once again until the city, so effectively 70% of your ride deals with absolutely minimal traffic. Once you've skirted the northern end of the Botanical Gardens you can just ride the 10min down to St Kilda along St Kilda Rd & Barkly St, which are well provisioned with both bike lanes & fellow cyclists. The dotted alternative route on the images is via Burnley St for 100m & then up Highett St, Richmond, before cutting down through the MCG/Rod Laver Arena precinct, which while involving some traffic I still found to be the safest way to add a shortcut in when I was strapped for time of a morning. Word of advice- steer clear of Chapel St altogether & Punt Rd/Hoddle St is 100% a no-go zone. The wonderful hazard of having many singletrack routes/options between Bulleen & Richmond is that you invariably chase a little too much ST on your way in & subsequently need to call on the 10-15min shortcut option to get to St Kilda Rd.
Load up on lights & wet weather gear & ride with plenty of commonsense & you won't have any dramas with the traffic. Definitely the safest cycling city in the country, but as with anywhere in the world there's still plenty of dickheads on the road.