Check the pots before reflowing anything, as that's the most likely culprit.
As for the adjustments and the retrace lines, that's a monitor adjustment (assuming it's just out of adjustment, and not another issue, as it could be either.)
When I adjust them, I turn the brightness and contrast all the way up, so you can see the retrace lines, and the dots in the center of the objects. Then dial the brightness back until the retrace lines and dots disappear. Then you can scale the contrast back to taste. I usually leave mine all the way up (which makes the vectors fatter, but brighter), but you can dial it back a little if you want a slightly sharper but dimmer picture.
The other pots on the game board and monitor are size (on the game board), and size linearity on the monitor. Use the test mode crosshatch screen to adjust these. The size pots on the game and monitor boards do basically the same thing, but you want the monitor ones dialed back enough such that you can get enough range with the game board pots (i.e., if the monitor is turned up too far, you may not be able to get the screen small enough with the game board pots).
The linearity pots are the trickier ones. These also adjust the size, but only at the edges of the screen, not the middle. These are used to correct the uniformness of the image across the tube. The trick to adjusting these is to make the crosshatch diamonds the same width across the screen, in both the horizontal and vertical directions. Look at the distances from corner to corner of the diamonds, and adjust accordingly. When you get it right, the lines should be pretty straight.
Adjusting linearity can be a little tricky (as it also affects the size as a result), so sometimes it takes a few iterations to get it close, and on some monitors you can't get it 100% perfect, just because everything is analog, including the tube. Patience is a virtue here. But you should be able to get it pretty decent, at least so that the objects don't change size as they move across the screen.