Unless you have a spending category that isn't covered by what you've listed, then there's really not much need to do so.
FICO wise you're fine with that setup; I might pickup another national bank card just because but that's more style points than anything else and future building. May want to consider a credit union or similar if you're just looking for a tradeline farm, they can be handy for other services later. Alternatively either a grocery card or a no-FTF card (international internet purchases) are often reasonable pickups... but other than fattening your file a bit which isn't going to have an immediate impact (later future buffer for AAoA, which you can get via Amex backdating anyway) it's seriously coloring at the margins
TLDR: you're doing more than fine already.
Yes, looks good. Possible addition to Rev's list depending on how much your property management payment is. If it is big enough, you may want to pay it through Bluebird or similar, and for that you want a card paying at least 1.5%, like Cap One quicksilver. You would also need a cooperative store, such as CVS, 7-11 or Walmart.
For $725, not worth the hassle.
Bluebird is an American Express product which does free bill pay (sending an electronic or paper check to your payee as required).
You "load" bluebird in various ways, including buying reload cards. With a suitable credit card, and access to stores that allow you to buy reloads with credit cards, it is a way of getting rewards on payments that do not accept credit cards. Without a 2% or more credit card, the time it would take on a $725 payment is probably not worth it.
Hahah I hear ya...don't we all.