As with many things in Taiwan, these rules are not applied (in this specific case it’s a good thing). My friends have a large black lab mix (around 40 kg), and they regularly take him on the MRT and local trains in a big dog stroller. He didn’t like it much at first but once he realised it meant he was going on an adventure he took to it pretty quickly. So far they’ve never had any problem, people usually think it’s cute. I’d avoid rush hour though 
I have a dog myself, a medium-sized Taiwanese rescue. I have a car and a scooter which she has learnt to ride on with me, so I’ve never had to take her on public transport. It’s very doable though.
As for where to live, anywhere near the river is good - as mentioned above, the riverside parks are great for runs with dogs: no cars or scooters, plenty of space, lots of interesting things to sniff at and even the occasional pack of stray dogs for your Siegfried to make friends with (not all of them are super dog-friendly, but up till now I have never encountered any really aggressive ones either - YMMV).
Another great option is anywhere near the hills & hiking trails: Beitou, Tianmu, Shilin, Neihu, Xiangshan etc. One of the great things about Taipei is that there are so many easy-to-get-to trails around the city. You can find an apartment right next to an MRT station and still be within 10-15 minutes walk from trail heads.
I myself live in Xinyi district, halfway between Xiangshan and the riverside park. It takes 5 minutes on my scooter to get to either, so I take my dog running in both areas daily. My apartment is tiny, which doesn’t matter to her because she gets all the off-leash exercise she needs. I would not trade my place for a bigger one outside of the city on her account, we have more than enough options around here for her to have a great dog life 
Here’s a facebook group you might want to join: Dog Friendly Hikers Taipei. They don’t organise hikes that often, but it’s a good place to get info on how and where to take your dog around Taipei.
