I would take it off next time you inspect. They probably won’t need feeding for a few months, if at all. If you leave it on, it is more dead space for them to heat (they keep the brood at 34 to 35C), and bees tend to do nefarious things to extras left on the hive. Things like building creative comb in unexpected places, or gumming stuff up unhelpfully with tons of propolis. 
I treat for mites when the counts say I need to treat or if I see DWV signs. Generally this is March, August and sometimes October. I use the sugar roll test and Gizmo device developed by UMN.
https://www.beelab.umn.edu/sites/beelab.umn.edu/files/varroa_brochure_final_print_2.23.17.pdf
https://www.kelleybees.com/search.html?Search=gizmo
What are you “fogging” with? I don’t like mineral oil, but Oxalic Acid is superb.
I presume you mean SHB (not SMB). I treat those with a hive tool (squish) - look for them around the edge of the inner cover when you first take it off. If there are lots, I use oil traps in the hive between 2 frames in the top box. Try not to squash bees - dead trapped bees are a feast for SHB larvae. Don’t throw wax etc from inspections on the ground - it attracts more SHB. I put wax into a ziplock bag and freeze it until I have enough to render. Keep a strong hive, and they will control the SHB for you.
Strong hive is the only way to deal with wax moths too. If you take frames off the hive, freeze them for 48 hours then wrap in burlap to keep insects out if you are going to store them. Freezing kills moth and SHB eggs and larvae.