I have a bullet cutter on my keychain, with me all the time. That way I always have one with me, just in case.

Mine is nice because you pop off the cutter, gently twist and push into the cap, and then after you're all done - by putting it back onto the keychain it pops out the piece of cap, cleaning it up for next time.

The "chop" filler you're talking about, I believe, is called a "short" filler. Long filler cigars are full leaves rolled into the cigar. Short filler is cut up tobacco, rolled into the cigar. Generally, those cigars are a bit cheaper, and not as nice as the long fillers. One of my favorite short filler cigars is the La Vieja Habana's made by Drew Estate. They're reasonably priced, and a not bad "yard gar" (for smoking while I do yard work).

I've got to imagine that using a punch will help the problem of picking small pieces of tobacco out of your mouth. I have that problem with just about every cigar I smoke, even long filler. The punch does tend to funnel the smoke, increasing the intensity of the cigar. I've also had problems, from time to time, of tar building up and plugging the hole.