Bought 2 Lufthansa tickets - one person can't travel - free seat next to me?

What if I still check in the second person - do i get an empty seat next to me, as i initially reserved the seat?

Not necessarily. Your boarding pass will say that you must board by a certain number of minutes before departure. Anybody who has not boarded at that time has their reservation cancelled, and their seat can be given away to a standby passenger. So you could end up with a standby passenger sitting next to you.

Will Lufthansa figure out that the second person did not board the airplane for the first leg towards Munich, and invalidate the second seat on the long-haul to Asia?

Yes, they will.

Will Lufthansa also invalidate the return ticket, if the 2nd person didn't even fly to Asia?

Yes, they will.

Could this cause myself any trouble?

Not really, though you will probably hear a lot of urgent announcements telling your companion to hurry up and board.

since the tickets are not refundable, i would at least like to make the most out of it.

Even for a non-refundable ticket, you can usually recover some value if you formally cancel ahead of time. You may be able to get the taxes and airport fees refunded, and for many types of tickets, you can apply part of the fare you paid to a future flight for the same person on the same airline (minus a "cancellation fee" or "change fee"). Read the fine print on your ticket, or call the airline. The airline wants to give you incentives to do this, because it gives them a better chance of being able to sell that seat again.

So it is probably a better idea to formally cancel your companion's ticket instead of checking them in and having them "no show".