I agree with the other posts.
Percussion caps usually split when they are fired.
This is a good thing as those that don't split and remain firmly on the nipple can be very hard to remove. Reloading a gun with the cap still clinging to the nipple is a PITA, usually requiring a knife to pry the stuck cap off.
Triple 7 is a hotter burning, more powerful muzzleloading powder. It was initially developed for the non-traditinal guns that shoot non-traditional projectiles that are shot without patches.
Burned patches are pretty common when they are shot with this powder. This is especially true if the patches are lubricated and allowed to dry before they are loaded.
As was mentioned, the Lyman guns barrels are protected by the factory with some unusually hard to remove grease. Until this grease is removed, accuracy will be poor.
Rifles with square groove rifling almost always has razor sharp edges as it comes from the factory.
Shooting 100-150 shots thru it will usually wear off the sharp edges making it much less likely that they will cut the cloth patch.
This "dulling" process can be speeded up by wrapping a small amount of #00 or #000 steel wool around the cleaning jag on a rod and running it up and down the bore.
The steel wool is about the same hardness as the barrel steel so it will dull up the sharp edges without actually changing the size of the bore and its rifling.
Another source of cut patches is the crown at the muzzle.
This is usually machined and deburred at the factory but the edges of the rifling and bore are often left very sharp.
These sharp edges can easily cut the patch while the patch and ball are being started into the muzzle.
To remove this potential problem, buy a piece of 220 or 320 grit wet/dry (black) sandpaper.
Tare off a 1" X 1" piece and lay it on the muzzle.
Use your thumb to push the center of the paper down into the bore and then rotate your hand/thumb back and forth while you slowly rotate the barrel.
It usually takes a few pieces of paper but when your done, all of the sharp edges will be nicely rounded off so they will never cut your patch again.
IMO, your patch material must be thick enough to be compressed into the grooves of the rifling.
Your rifling grooves are .010 deep on your GPR so one might think that some .012 thick patches could fill it tightly but remember, your using a .490 diameter ball.
Adding .490 + .012 (patch on one side of the ball) + .012 (patch on the other side of the ball) gives a uncompressed patch and ball diameter of .514.
Adding the .500 diameter bore size with the .010 depth of the rifling gives a size of .500 + .010 (rifling on one side of the ball) + .010 (rifling on the other side of the ball) results in a rifling groove diameter of .520.
Obviously, the .514 diameter patch & ball won't seal a .520 diameter groove.
Even using good old black powder, the resulting gap will allow the flame from the powder charge to blow past the patch and burn it badly.
Add to this the use of T7 and it is much worse.
* To sum up, dull up your rifling edges and the sharp corners at the muzzle.
* Use the .015 thick patches with your .490 diameter ball.
* Use slightly damp patches, not dry patches.
* Try putting some non-combustable filler over the powder charge to keep the heat away from your patch. 10-20 grains (by volume) of corn meal or Cream of Wheat will help.
PS: If you switch from T7 to real black powder or Pyrodex and use the tight ball/patch combination, fillers won't be needed to prevent burning the patch.
Oh, the shot patch will always look tattered around the edges. If the important part where the ball meets the bore isn't torn or burned, the ball/patch/powder combination is good to go.