Does mining activity only exist if there are bitcoin transactions? Bitcoin's mining only exists, if there are some transactions, right?
If nobody trades bitcoins, the miners don't have any transactions to validate, and so the number of bitcoins in circulation doesn't increase?
This is not correct.  There is no requirement that a block contain any transactions at all (other than the single transaction indicating what address should be credited with the block reward, currently 25 BTC).  Miners can mine whether or not there are any transactions on the network for them to include.

See also What is the smallest block ever mined?