If you hook up the Mac via ethernet or thunderbolt-2 ethernet adapter, it could run faster if it supports it.
Mountain LIon will only install on Macs releases prior to Mountain Lion, and via Internet recovery if you are lucky and able to install via it by resetting the Mac's date/time for its year. Please note, since AppleID is only available Sierra and later, trying to get it via the App Store is probably extremely difficult at this point unless it was already downloaded by you at a point in the past.
A few well rated places may also have Mountain Lion on USB drive on eBay.
Keep in mind Mountain Lion, Mavericks, Yosemite and El Capitan will all run on Mountain Lion compatible Macs. The only reason to run Mountain Lion is for software not compatible with later OSes.