Most geldings don't bother to mount mares but a few will and that's why most livery yards separate the sexes into separate turnout herds..
If you don't know the history of the gelding then it is possible he is a rig (castration not completely effective and some testicular tissue still producing hormones internally) in which case he might show other riggy/stallionish behaviours such as herding, aggressive territorial guarding etc and would be fertile which is why your other friend is worried.
I had a late cut gelding (cut at 6) who once penetrated a mare about 3 months after castration when first allowed to run with my mares and geldings - he had a strange expression which seemed like confusion, he'd been desperate to do this all his life and suddenly got the chance and it wasn't meeting expectations
He never did it again but was quite a herding type who loved being the big man and boss of other geldings for life (I had him to 17). None of my others has ever done it in over 20 years so I'd be wary and also separate them particularly if he is shod - whether fertile or just annoying, shoes can do alot of bruising particularly if he flails about while up there..
Roxy the lipcurling you describe is called Flehmen, the way a horse intensifies strange or interesting smells by enclosing the sample within the nostril area and raising the head. All equines do it but it is particularly strong behaviour in stallions as they need to know when a mare is releasing hormones that tell him they are starting to come into season. Getting it wrong with a mare's cycle can be a painful mistake
so they're constantly testing and comparing internal notes to get the best chance of safely approaching and breeding.. Youngsters also do it a lot as so many smells are new and interesting to them