You can fill the stock with pretty much anything: an old tshirt or shop rags, spray foam insulation, just make sure to do it in small layers to make sure it dries right, high density foam cut to fit are a few ways you can. Also recommend you get some 5mm flat washers to install in the front stock screws between the star washers and the stock. Those star washers will chew it up over time and the flat washers will help stop that. You can use Loctite, but if you can get it, Vibra Tite VC-3 works better. It lets you make adjustments to the screws if you need to without having to remove, clean and reapply it.Yes the longer screw, usually an m3 x 8 will do the trick, plus polishing and light lube of everything. Then just keep shooting and use will help finish smoothing everything out more.Define longer ranges and I guess what you use the rifle for. Anything you do, be it a Vortex gas ram with higher pressure or a stronger spring, will make the rifle harder to cock and recoil harder which will make those nice groups suffer. It will also put more strain on every part of the rifle and things will start to break faster from that harder recoil.
For accurasy at longer distance i would recomend go up a bit higher in pelletweight, all my hatsans love heavier pellets, in 177 try jsb 10.34 or the 13gr ones
I wanted to add some weight to my stock for better balance & to help quieten it some. I used " Plumber's Putty" & put it in a zip lock freezer bag. It does not harden & you can mold it to fit. You can add or remove some of it to see how it does for you.
Another thought would be what scope you're using. Not a clue how they sell them in Europe, but here in the US they come with Optima 3-9x32 AO scopes that have a bad habit of not lasting more than a few hundred shots and need to be replaced.
I have a Hatsan Edge and a Dominator and have taken them both apart and put back together. My Dominator has a spring but the Edge has a gas piston- the Type3 Vortex piston. I replaced the spring in my Dominator with a 0.128 inch diameter wire spring with 35 coils and 0.790 outer diameter that I bought from Vortek Products. I would suspect that the same spring would work well in the Edge. I wouldn't go larger in wire diameter though. Or,a spring with wire diameter of 0.125 inch would probably be very smooth firing cycle which would make for accurate shooting. Those are my recommendations to get you started. For awhile though, I'd recommend just leaving the current spring in and use the rifle for awhile. Some of the other components will "settle in". Be prepared for the pivot bolt to loosen up on you. That's an important one to locktite down. Until I locktited mine, the barrel pivot could loosen up in as few as 5 shots.For the 0.125 and 0.128 diameter wires, you could experiment with 35 or 40 coils for the spring. Experimenting with the outer diameter of the spring is worth trying as well. If you do swap springs out, please let us know what your results are. Chronometer results for different pellet weights is always interesting. The way the end plug is designed on the Edge, it appeared to me that swapping out the vortex piston would be very doable. However, I like the Edge with a gas piston. So, I'm leaving mine the way it is.
My mistake. You wrote Striker Edge and my brain only saw "Edge". I often get that confused. I have a Hatsan Edge not a Striker Edge. So, I can't recommend spring measurements. Sorry.I'd recommend getting a digital vernier caliper if you don't have one. You can use it to measure the wire diameter of the OEM spring and the outer and inner diameter of the spring. Then use that info to select a spring to buy that you might want to use. If you don't have a chronometer, that's also a worthy purchase if you are wanting to experiment with power plants in your rifle. Using a chronometer is really the only way to know what your changes are making the rifle do. You'll definitely want to use the chronometer to measure fps before you make any changes. That'll be your baseline to compare all of your other changes to.
The Edge, Striker and 1000X rifles except for stocks are the same rifle with the same power plant whether it be spring piston or Vortex gas ram. I've owned every caliber of the Edge rifles and still have all but the .177 that, before I blew the ram in it tinkering, had been tuned and some customizing done to the stock and muzzle break. I still have the stock and some odd parts and pieces, the rest has been used in other rifles or donated to other members.Because I own quite a few Hawke scopes (25 at last count) don't think you can go wrong with one. All of mine, 2 in .25 and one in .22 all have Hawke Vantage 3-9x40 AO IR scopes on them and I have no complaints. I also have a .25 cal 1000X that has the above mentioned Hammers 3-9x40 AO and no complaints there either. I couldn't find the link for the thread on how I went about it, but here's what my .177 Edge looked like after I got done tinkering: