separating proof of work from mining possible? I (think to) understand that proof-of-work is the process through which transactions are verified and the health of the network is being ensured.

I also (think to) understand miners get bitcoins for successfully verifying transactions as a reward. 

But can the actual mining of coins (in a fork) be separated of proof-of-work in the context of transaction verification?

Like for example a hardware device generating coins with a different ruleset (e.g. let's say for simplicity: a car drives 100miles --> generates one coin). If that would be possible, would transaction verification still be possible or is it intrinsically linked and in itself one process?

Hope I am making myself understood.
Never mind if the question is making a fool of me.
All nodes verify transactions in Bitcoin. This helps propagate transactions in the network fast and given a majority of honest nodes you propagate valid transactions faster.

Given that anyone can verify transactions it will not be difficult to have some dishonest nodes to verify 2 of my transactions spending the same bitcoins (double-spend). We do not want that, so we need to make it difficult to publish (verified) transactions to the network. Thus we have proof-of-work/mining** and some of the above nodes also mine. 

So the introduction of PoW was so that publishing*** (verified) transactions is safe from double-spends. And to incentivize the process bitcoins are rewarded. 

Your example with the car (I assume that it could work although I see several issues :)) could work as PoW since it requires effort. But you need to prove your effort for something. In Bitcoin it is for adding (and verifying) new transactions to the blockchain. What will it do in your example? And how would you safeguard from double-spending a car-coin?

** Also works as an anti-spam measure.

*** The reward is for publishing a block (set of transactions) in the blockchain not just for verifying. Even though many others will verify only the nodes that prove their work are allowed to add a new block in the blockchain.