Pretty much nailed it all.

You need the wheelset, spacer, presta tubes, and cassete.
You dont need a new chain. But, its often recommended to replace the chain when you replace the cassette. Not required, but often done. A 7sp kmc brand chain is probably $8 online, so if the current chain is well used, it wouldnt hurt and is a small cost. If the current chain is relatively new or lightly used, leaving it is fine.

You dont need new tires. Get new tires if you want, but your current tires will fit fine.
Get 3 tubes at least so you have an extra when needed.

I would bet that the new wheelset is heavier as you suggest. Its a 40spoke wheelset using a rim that is not a lightweight rim.
As for how itll ride...i wouldnbet itllnride with less flex. The stock rims are flexible. I am 6'5 230# and can torque those stock rear wheels very easily. This lateral twisting is in part why the wheelset isnt great.
The velomine wheelset is 40 spokes and a robust rim. Based only on those 2 things, i would think it will flex less.

I find that tires change the feel of a ride more than rims. My experience is admittedly limited, but ive ridden the same tires on a few different wheelsets and the ride wasn't really different between wheels.
Change the tire from a cheap thick heavy tire to a light supple folding tire and i notice a big difference in how the bike feels and rides.