What heart pieces are unavoidable

Fill the bar about half full at the start, then wait. The guy will give you a final warning 10 seconds before the auction ends (I believe he literally says something about 10 seconds). Mash like mad at this point, and generally they'll never manage to get a bid in after you. Once you've got the timing down you can easily get the bid in with 3~ seconds left, and I've not once had someone slip in a bid after that time. If there's less than 5 seconds left there's zero need to stun them, just add a single rupee.

If the auction bar accidentally fills early, do NOT place a bid. Attempt to pay the current bid and your bid will be rejected and your bar starts half full instead of empty. Unless you make this mistake extremely close to the end, you'll have plenty of time to mash A to fill the bar again. Also note that bringing up the bid UI will show you the exact time remaining, and note that the infuriating dialog of other auctioneers does not eat into the time.

You can stun the other auctioneers with a significantly higher bid too (not sure how much, 50 rupees?) and prevent them from bidding, but in my experience if you do this too early it's useless, and if you do this too late you might as well just wait for the 10 second warning and you should pay the same or less as you would have by bumping the price up considerably.

It's been a while since I installed the game on my Wii U and I can't check right now, but I'm quite sure the size is around 2.5 GB plus/minus a few MB due to compression and such.