Airside transit is easy - there is a shuttle train. CX is terminal 1, OZ is terminal 2. EVA is also terminal 2 if you wanted to fly HKG-TPE and arrive in terminal 2, but really it doesn't matter.

Assuming you are on one ticket, you should be able to check the luggage straight through. In this case, 1 hr 45mins is bags of time. If you have to clear customs/immigration then with a terminal change, 1hr 45mins is cutting it a too fine IMHO and I would allow a bit longer. TPE is efficient, but in rare cases immigration queues can keep you waiting 30mins or so. Landside transit takes a bit longer.

I'm no HKG expert, but you need the Airport Express. I believe you can drop luggage up to 24 hours before if it suits you. Stops at Central, Kowloon and then on to the airport with a brief stop at Tsing Yi. Central and Kowloon are so close that the extra journey time from Central is not significant. I prefer Central which has better MTR connections. Perhaps hotels are cheaper near Kowloon station though.