Love, love, love paper piecing! I don't know who your pattern is by, but I do a lot of Judy Niemeyer and find her instructions can sometimes be overcomplicated for new pp-ers.
Yes - stitch past the line and if you haven't already done so, reduce your stitch length to around 1.5 - this will make tearing off the paper much easier. Much of this will be cut off when you trim and depending on how your pattern folds and crosses, the 'extra' will loosen. If you haven't reduced your stitch length it may loosen too much.
Also, when I tear off my paper, I tear it it half the opposite way, down to the stitch line and remove the two halves separately - less stress on the seams. Another tip is to run a dampened cotton bud along the paper seams.
With regards to chaining or completing each block, you'll probably get a 'feel' for what suits you and the pattern. I've done both, sometimes getting into a 'zone', lol. I chain a lot,
but my machine has auto-pivot where the foot lifts a little as soon as I stop sewing which makes it easy for me. When blocks are all the same it usually lends itself to the freezer paper method, but that's a whole other kettle of fish!
Templates - I often make my own template out of freezer paper when there's a lot of pieces the same size. I usually cut it a bit more generously if needed - 'fickering' (a good Scottish word for you, lol) with bits of fabric to get it to fit ain't my thing. It's less time-consuming having to cut off any extra that it is turning pieces this way and then having to unpick pieces that are too short (remember those
tiny stitches.....).
You can always cut smaller, it's bigger that's the problem
. That way you can iron it on to the top of pile of likely looking scraps and rotary cut a stack, remembering to cut the right side of your fabric to fit. (Ah - batiks heaven sent for paper piecers....).
Phew! That's my tuppence-worth. Happy piecing!