What is the best practice to deal with inactive channels?
Do they have an impact on my node's 'reputation' somehow, should I close them after a few days of being inactive?
What happens if channel funding fees are involved?
You should consider the channel inactive, if after 2 weeks, the other channel does not broadcast a
channel_update. At this point, you should try to close the channel cooperatively, and if this fails, broadcast the most recent commitment transaction.
If the channel is active, but just not routing payments, then you should try to figure out why that might be the case. Is it due to liquidity issues on the channel itself, or the other party's liquidity issues, or perhaps liquidity issues in your own channels?
For example, if the other party has no other channels, it will only ever route payments that it makes. If you have only one channel, or no liquidity, then you will never route payments other than your own through the channel.
If you feel that the party you're connected to is not maintaining their node and it is negatively affecting you, then you should attempt a cooperative close, and use the funds to open a new channel to a different peer.
As far as I'm aware, the current implementations do not determine the reputation of their peers, except obvious malicious actions which result in blacklisting. It is currently up to you to investigate and determine whether or not peers are useful.
This will likely change in future software updates, when we have more information about how people are routing payments and what is considered good or bad behavior. It would be premature to make too many assumptions at this stage.
When you close a channel, the fees depend on whether it is closed cooperatively or unilaterally. A cooperative close is a regular, small transaction, and the feerate is haggled, until both parties agree on the fee.
In the unilateral close, the fee depends on how many HTLCs are in flight, and the most recent feerate which has been agreed. The base transaction weight is 724 bytes if there are no HTLCs.