Hi Adam,
I can answer some of your questions. Also, "Padova is on a different train line than Verona from Venice. I’m pretty sure you have to train into Venice to get to Padua. " isn't correct. You can take the train from Verona or Padova without going to Venice. The search below on www.trenitalia.com shows that it takes 42 minutes.
https://www.lefrecce.it/B2CWeb/search.do?parameter=searchOutputViewer&cFID=2g32GlvwEqw7
I've attended an opera at La Fenice. Definitely go if there's an opera while you're there. Buy tickets on-line soon. The room has box seats in an arch arrangement, so make sure you're not in a partial view seat. If I remember correctly, the two chairs in the back of the box seats were higher than our front row chairs.
Speaking of opera, the Roman arena in Verona is a fantastic location to attend an opera! They won't be on during the time you're there, but there might be a concert or similar playing at the arena.
I've traveled with my husband and also alone in Italy. I don't know of any option for eating with locals, but another option is taking a cooking class. Last time I was in Verona, I took a cooking class with Veronality. Your chef will share info about the town, culture, local food, etc. I also took their city bike tour. I had been to Verona a couple of times previously, but it was fun to learn new info about Verona in this fun method.
I've been to all of the cities you're mentioning. Personally, I wouldn't try to go to all of them or at least keep your plans flexible. You waste time in sitting on trains instead of walking around seeing the sites. For Verona, you can buy a Verona Card which has a list of sites to see. I would recommend it. You're going to have a jet lag day in Verona, so walking around seeing sites with the card is a good option for your first full day.
Venice is so unique, I definitely would not plan any day trips from it, except seeing Burano or Murano if you wish.
What I wish I knew before we went the first time: hmmm, maybe knowing that dinner starts later in the evening than we expected, and the later you can go, the more you will be eating with a room of local people instead of tourists. Try to adjust your time to their later dinners.
Now it's quirky & just Italian, but the first time there I was surprised that there's no lines in Italy. Everyone just kind of crowds into a spot and it works! So if you want to be on the boat, don't pause too long. ; )
Have a great time!