Installing insight blockchain explorer - is bitcore mandatory? What I have: bitcoind on Linux running.
What I want: to have insight api+ui local website running.

Earlier it was possible to add insight querying your bitcoind. Now it seems they made some bitcore base for insight.

If I install bitcore is it gonna replace bitcoind? How do they interact?

What's the best way to have local insight explorer running now?
Bitcore is a javascript library for interacting with Bitcoin. Bitcore is built on top of bitcoind and doesn't replace it.

Insight is probably the simplest way to run explorer-style queries if you've already got bitcoind set up.

Alternatively: I believe btcd and Toshi offer similar functionality, but with different implementations than bitcoind.