I think is over, we have almost 14bb.

I would limp/fold, and in case he checks, min bet flop.

With not a lot of tendencies to go off, I'd probably push for a raise/fold to steal, Q7 has not got a lot of post-flop playability especially OOP. I definitely don't think we should be pressure-shoving Qs7c here.

BB VPIP is 30%: 55+,A2s+,K5s+,Q7s+,J8s+,T8s+,98s,A7o+,A5o,K9o+,Q9o+,J9o+,T9o

You're only getting around 35% equity. What are his fold stats? You've also got to think what tendencies he's got on you, are you opening wide a lot to steal?

Based on Nash equilibrium ranges, you should be shoving 51.1% {22+ Ax K2s+ K3o+ Q3s+ Q8o+ J5s+ J9o+ T6s+ T8o+ 95s+ 98o 85s+ 87o 75s+ 64s+ 54s} here, therefore Q7o is just a dash too loose.

However, if you suspect villain will be calling tighter than what nash equilibrium dictates (which might easily be the case), then shoving Q7o in this spot will be profitable.

Also, while VPIP and PFR are decent indicators of a player's general style, they don't allow for specific insight into villain's calling/shoving ranges in spots such as the one in question, so you can largely disregard them when deciding whether a shove will be profitable or not.