It will do no real harm to use the old heat sink, but a square aluminium one would be better, and a copper one would be best. Weigh the cost. DO NOT BUY A HEATSINK THAT IS TALLER THAN THE ONE YOU HAVE - it must fit under the keyboard.
If the heatsink fell off - that would explain why it keeps overheating.
I would be happier replacing it with a copper or brass heatsink (better than aluminium one). A square one to cover the center of the chip, because that's where the die of the chip is. You can use more than one to cover the area, but a single one about 40mm X 40mm offers a fairly good fit.
As for glue to use, look for "thermal conductive adhesive". Germany makes some of the best you can get. There must be shops there selling the glue to overclock enthusiasts. Also, 3D printer shops often stock the products, (because they are used to glue heatsinks to motor controller boards in 3D printers to help keep them cool). Don't be in a hurry, shop around for the best thermal conductivity, low viscosity (most "dribbly"). You will have to make sure there are no air gaps in the join,
As for a "cheap" VGA connection - all the cheap ones rely on the monitor being able to handle Amiga sync rates, which most monitors do not. The expensive external solutions convert the slower signals to a true VGA standard, and the internal ones are even more expensive. A scan doubler works to a point, but will cause a "torn" display on a fast 50Hz game.
I got no answer for you on that one. I could well be wrong too, perhaps there is a cheap scart-VGA adapter on amazon or similar, that works in all Amiga screen modes,
EDIT: What might work is a cheap scart-HDMI converter, then a cheap HDMI-VGA converter. I have tried the latter, they do work, but generaly with high resolution VGA monitors, 17 inch display or larger (must have over 1,000 vertical lines, 1280 X 1024). I don't have any experience with cheap scart-HDMI converters.