Which stable and modern custom ROM can I install on Galaxy Tab 10.1 (GT-P7500)?
It Looks like there has never been an official supported stable build. That being said, you can try some of the nightly builds and hope they're stable.

Alternatively, you could have a look at some other ROMs such as ColorosOS, ParanoidAndroid, SlimROM (Check the compatibility, i've never used these on a tablet).