I can't comment on UA or Lufthansa, but I recently flew BDB-JFK-PRG-LCA on DL (PRG-LCA was on Czech Air - codeshare) with my cat in the cabin and my cat and carrier were never weighed and her paperwork was never examined by any airline employee.

Please do not sedate your cats when traveling by air. I feel it is too risky. I found that keeping my hand in the carrier to pet the cat or just rest my hand on her kept her quiet. She did get stressed a bit when we had to get off the commuter plane in JFK - way too much airplane noise since we did not use a jetway to disembark and I had to wait for my pink-tagged bag. But mostly she slept.

Do you have EU pet passports? Makes things very easy when traveling within Europe. If not, you might want to consider getting them once you are in Paris.