Sadly a friend of mine is absent today, so I couldn’t get his build as a referrence (and yes, he’s my partner in HC multiplayer), so I’ll try to be brief.
First of, why fire and cold? I mean, Inquisitor and Demolitionist have an obvious fire and lightning synergy, so why deliberately ignore lightning? Ofcourse you can get Chilling Rounds if you want to.
Second, why do you need Word of Pain? It’s a waste for the gunslinger, or do you build caster? Saved points can be invested in a more crucial stuff.
Third, Inquisitor Seal is a superb defensive ability - max it. It’ll allow you to stand still and shoot instead of running hectically. It can pretty much carry you through normal if your other defenses do not suck. Seal makes inqusitor-based gunslingers quite safe before it’s end game, and even then it’s very useful.
Put a point into Null Field and max Arcane Empowerment as well.
Next, you don’t need Ranged Expertise maxed, because you’re not a pierce build. Put 2 points for some cheap attack speed.
Put 11 points into Vindictive Flame (sweet spot) and 10 points into Thermite Mines (sweet spot)
The way I see it skill build will be something like that:
https://www.grimtools.com/calc/nZoD4yzN
Usually there’s opportunity to take more stuff due to +skills that items have, but since you don’t have any there’s nothing to discuss. As a general rule you don’t need more than 11 in Vindictive Flame, more than 10 in Thermite Mines and more than 12 in Deadly Aim, so if you happen to gain more remove hard points from skills and reinvest them elsewhere. Many people fancy extra points in Blast Shield, but I think 12/12 is fine. It’s not your main method of defense, more like a safety belt.
Word of Renewal is among the things you need to consider if you happen to get some spare points. Vigor is esp. good for cc resists, which are hard to come by.
As for levelling there are a great number of strategies. E.g. open Inquisitor, max Ranged Expertise and put points into mastery bar. Then at level 10 get Demolitionist and respec Inquisitor leaving only 1 pt in Ranged Expertise and reinvest into Fire Strike and Explosive Strike.