Do I need to sand between coats of varnish

When you say you can see the brush marks, do you mean that they're actually irregularly surfaced? I mean - are we talking just a visual effect or an actual difference in the depth of the paint?

If the latter, do a skim coat with lightweight joint compound and when it dries, either lightly sand OR smooth with a large, slightly damp sponge, then apply primer.

As for your own brush strokes - orient the roller so the roller bar leads your motion. If you're moving right, put the bar on the right. If you're moving left, put the bar on the left. The most pressure will be on the roller bar side, so keeping that side on the "front" of your motion will mean the lighter touch side (the back) will smooth out the uneven strokes as you pass over it again.

Now if its not a real issue with texture, just that your seams are obvious - then just prime it again with a good HEAVY primer, using the proper brush techniques, and you'll be fine. :)

You shouldn't need to strip unless you scratched through to wood.

The problem with polyurethanes is that they are hard films that don't blend with the previous coat. Other finishes will 'melt into" the previous one (lacquer, shellac, tung oil)

Your sanding, @220, was too aggressive. Intercoat sands should be at 320 - 400.

You will have to sand back to a common level that eliminates the scratches and lines. If you leave part of one coat and part of another coat, there will be a "witness line" . You can wipe down the area after sanding with a soft-mineral spirits cloth and look for the lines.

Be sure you are allowing enough time between coats with the poly.