Most female livebearers are pretty much permanently pregnant if there are males in the tank so I would say that's why they look fat, and as you've guessed, any fry are being eaten by the adult fish in the tank.
The API Master Test kit is what most of us use. The only things it omits are gH and kH which can be bought separately if required. If you do go down the RO route, you'll also need a TDS meter to check your RO so you can tell when it's time to replace the membrane and to measure how much minerals you're adding.
Does anyone gave any recommendations for the best water testing kit for a tropical tank?
The API Freshwater Master Test Kit sounds like a good one - but us it adequate? The last comment in the review I read said: “this kit is usually enough to monitor and maintain a simple tank” which actually makes me wonder if I need something more comprehensive!
Try treating with Esha 2000. I've had success treating a honey gourami for dropsy with that in the past. However, from what I can see the fish is starting to pinecone (scales lifting upright away from the body) and once it gets to that stage it's usually too late for any treatment to work.
SORRY - just saw your additional post.
Use two nets to catch fish, hold one in position and use the other to guide him into it.