I cannot help you with all your locations but can throw out a couple places.
However, I will say you may need a vacation when you return from your vacation… you guys will be covering some territory.

In Florence we stayed at the https://www.sangallopalace.com/
Very nice hotel and reasonable. It is not in the thick of things. About 12 blocks to the center of town, but far enough away to escape the mass of tourist that over take the city. Great neighborhood restaurants around the corner. There is also parking nearby, which is a consideration in Florence if you are driving.

For the Cinque, which is an amazing place, if budget is a consideration you may want to consider La Sepzia. We stayed at My One hotel, but now under new management and is called the CDH Hotel La Spezia. Great Location! And once again under your budget. You are two blocks from the ferry to the Cinque. However, I would not skip Lerici or Portovenere when you are in that area. All three places are a very short ferry ride from La Spezia. La Spezia is a real town and not very touristy, so at night you will get to experience the Paseo. In the evening the children come out and play in the square, grandparents gather and watch them while the parents sit in the cafes enjoying adult time. Then the family will walk together through the streets and alleys for dinner which is served promptly at 9 pm. It is not as romantic as the Cinque... but not sure anything compares.

You are going to have sooooooooooooo much fun!