Help with cassette, worn or not?
By looking on the shape of the tooth on the biggest cog, it looks worn out and a time for a replacement. Grinding feeling on a clean chain is always a sign of wear.

Also worn-out cassette will will wear out chains much quicker. And you already feel that - second chain got only 50% of usage from the first chain. 

I would just replace the cassette, before the new chain is also stretched out. 

p.s. you are far too precise with your mileage!