Sounds like your motor is failing the visual inspection.
I think they're probably looking for the PCV and line that picks off the right /left front valve cover.
Lots of folks install breathers instead.
JPC- 2011+ GT/Boss Breather Package - Justin's Performance Center
5.0 DOHC Boss and Coyote Cam Cover Breather Kit - Scram Speed
This guide may be of help.
https://lmr.com/products/How-To-Inst...011-14-Mustang
On the battery. Any of the AGM (absorbent glass mat) should do you fine. I like them as they're spill proof. I'm partial to Odessey (not Optima) and Exide batteries. To be honest you may get a lot of input as its close to asking what
oil you run?
This chart will help you find a good fit.
BCI Battery Group Size Chart