The Delta Gold I'd go ahead and close if the benefits/AF/limit aren't what you want. The BCE if you could I would hold off on closing for at 6-12 months so you don't give AMEX any ideas on a possible SUB app denial. It has no AF and it can keep til your AMEX card lineup is more to your liking before closing it as well.
My concern with apping for the Delta Plat is that you're still going for your mortgage, right?
If that's the case then hold off on the app, close the Delta Gold card so you don't get charged the AF, hold off on the BCE (just SD it), and when the elevated offers come around again whether it's increased miles, MQMs, or both then decide if the mortgage hasn't closed yet if it's worth it. Personally, I'd wait til the sale is final but that's just me. A new home for your family should be your priority atm over a SUB/companion certificate.
For credit reporting purposes yes but not with the bank/creditor. I feel like I had to make that distinction just to be safe.