The fill valve looks like a custom Fluidmaster. If so, you can replace the working seal for about $1. A second problem is that the refill hose is tucked into the overflow tube...THIS IS NOT GOOD!! It needs to be above, held in place by the (originally) supplied clip otherwise, it can cause the tank to siphon, and waste water plus provide a potential contamination situation. Most newer valves are designed to prevent that cross-contamination, but it's best not to push your luck or waste water!
The flapper with a float is required to ensure the designed amount of water leaves the tank and it closes when it should. There are some that are adjustable, and with a little fiddling, you should be able to make that work. To find an OEM replacement, you may need to go to a plumbing store. If you get a flapper that doesn't close at the right time, you'll either use more water, or not enough. The former won't affect performance, only economy. The later is a problem that will often result in two flushes....the first to refill the bowl, the second to actually flush.