The by-pass valve is designed to keep the water temp in the heater matrix from getting too hot and blowing it. The metal part of the bypass valve has a small thermostat in it, the plastic bit a one way valve. These often fail so they shut off the water supply to the matrix completely when the engine gets up to temp so you get heat for the first few miles then nothing.
When I got my 8v last year the temp gauge was reading very low but the heater worked quite well. I replaced the stat with an oem unit (don't trust any others) so the temp gauge now sits just below half way but it killed the heater after 5 miles because the hotter running temp shut the bypass valve. I have since 'modified' the bypass valve and I now have a toasty warm heater.
Your rad is clearly working fine or the engine would overheat. I would try a new oem stat from a vw dealer first to see if that sorts the gauge reading, then a new temp gauge sender. If you end up with no heater then sort the bypass unit out, if you had heat before you stuffed cardboard in front of the rad then the heater matrix is OK. Take the cardboard out to get your heater working again while you sort out the other bits. Running at a lower temp for a while will not be a problem, the oil will not get up to proper temp if the coolant temp is too low so the problems are probably related.
Was the new thermostat you fitted a genuine VW unit?
