How the difficulty of the network affect the average block mining time if we have static hash power? Let's suppose that I have 500 KH/s of hash power overall from my mining rig. I'm about to start mining coin X and its network difficulty is Y. Current hashrate of the whole network is 100 KH/s and calculations are showing that average time of mining one block by my rig is about 1 minute.

I've connected my rig to mine this coin and mined 10 blocks within 10 minutes and let's suppose that difficulty gradually was doubled and became 2Y. Will my rig have to mine next block for 2 minutes instead of the last one?
The rate of you finding blocks solely depends on your own hashrate and the network difficulty, nothing else.

The overall network hashrate influences the difficulty with a delay, which on its turn influences your rate of block finding. However, that's the only effect.