I load many thousands of rounds of 44-40 every year for cowboy shooting, and have been doing so for a long time. My first piece of advice -- forget about superlight bullets. The 44-40 works best with bullets in the 200-225 grain range, with 180 being the absolute minimum. Anything smaller than that is essentially a washer and not a bullet. I use 205's, myself. If you find the recoil from a lightly loaded 44-40 with 200 grain bullets to be too much for you, change to something like .38 or .32.
As for bullet impact -- yes, bullet weight plays a major role in point of impact with revolvers. Heavier bullets impact higher than lighter bullets. As 308win was explaining, a heavier bullet impacts higher for two reasons. First, it generates more recoil force. Second, the bullet accelerates slower. Combines, that increased means the barrel climbs more between the time you pull the trigger and the bullet leaves the barrel. I did some experimentation with my cowboy revolvers about a year ago and discovered that a 240 grain bullet impacted a good three inches higher than a 200 grain bullet at 7 yards. That's a difference that is big enough to notice even in cowboy shooting.
Fixed sight revolvers (like most cowboy guns) are typically designed with high blades that result in shots printing low, so you can file the front sight blade down to bring point of impact up to point of aim with your preferred load. Once you do that, though, changing your load much with result in revolvers that no longer shoot to point of aim. A small change in propellant charge and velocity won't have much effect, but even small changes in bullet weight can cause major shifts in point of impact.
A good "mild" smokeless load for 44-40 is a 200-205 grain bullet over 4.3 grains of Clays. It should produce around 600-700 fps depending on barrel length.
Of course, the 44-40 is one of the all-time great black powder cartridges, and really should be stuffed full of the Holy Black propellant. My standard cowboy action load is a 205 grain "Big Lube" bullet lubed with 50/50 over 35 grains of Goex fffg and lit by a WLP primer. It gives a nice satisfying boom instead of the wimpy "crack" of smokeless powder.
One last point -- 44-40 can be tricky to load for several reasons. The biggest is the bullet size issue. Originally, 44-40 firearms had .425-.426 bores and used .426-427 bullets. Factory 44-40 ammo is still loaded with 426-427 bullets. A few decades ago, however, many gun manufacturers started using .44 Special/Magnum barrels on their "44-40" firearms. Those barrels are .4295, and do best with .429-430 bullets (.430 in lead). Uberti, for instance, has used nothing but .4295 barrels for at least the last several decades. The only way to be sure is to slug your barrels. If you're not sure, .428 lead bullets are a decent compromise, but don't go over .427 in jacketed without slugging.
44-40 also has very thin case mouths that can be prone to crumpling if you over-crimp. Take your time to set your dies up carefully.
