There's a lot of good advice here.
I have an HW30s and did the Vortek tune. Didn't gain any power, so that was marketing BS, but it is a LITTLE bit smoother. Reading above at Bear of Grayling's advice, that is a good enough reason to do the tune. If you do it sooner rather than later, you won't have to break the gun in twice.
If you DON'T do the tune, it will be fine for many thousands of shots before it wears out, so I guess there's no rush. A nice springer like the HW50s is a real pleasure to shoot, and you were smart to not go for a magnum. The lower power ones really hold the interest longer. I think the HW50s was a good choice of power level, too: it's significantly more powerful than the HW30s, so it can be used on small game out to 25 yards or so, but not so powerful that it is going to be really hold-sensitive or heavy or hard to cock.
Break-in pellets can be cheap; I used Crosman Premiers in the past, but I hear they're on the way out. Maybe Excite Hammer (which are a RWS Superdome look-alike).
With the Hawke Airmax scope, you won't need to worry about it breaking, especially under the light recoil of this gun. No need for a Dampa mount and its expense or complexity. Heck, in time, you might end up putting that scope on something that kicks harder and put something lighter duty on the HW50s.
Yes, you should Loctite the stock screws. The two at the pivot and the one big one at the back of the trigger guard. If you don't, you will find your zero moving and you'll have to tighten them again and again. I use blue Loctite or the equivalent from the auto parts store.
As for cleaning the barrel, the easiest thing is a length of stiff weedeater line and patches soaked in GooGone or something similar. The guns have a thick grease (or cosmoline) in the barrel to keep it from rusting in storage and it'll take a few runs to get it out. Push one saturated patch through, wait a couple minutes, then push a dry one through. Repeat until the dry one comes out clean. If you find that's taking too long, you can get a Bore Snake; they make one that's specifically for airguns. (airguns use a softer steel in the barrel than powderburners, so you don't want to use anything too abrasive, like a bore brush with a steel rod)