Does Lufthansa weigh your carry on luggage?

Yes. According to Carry-on baggage rules at Lufthansa

For a smooth boarding procedure, more stowage space on board and a punctual departure, it is essential that your carry-on baggage corresponds to the regulations. That is why we check your carry-on baggage against the permitted dimensions, quantity and weight at departure airports.

And from a detailed Trip Advisor post about Lufthansa carry-on baggage

My experience - YES. It's weighed at check in and can be weighed at the gate for passengers connecting from a different airline to LH. And I have seen people sent away to consolidate their carry on weight down.

Here's what I've done in some cases (chortle chortle). Note you are also allowed a smaller personal item (e.g. laptop bag). I've had cases where I have moved a heavier item (e.g. laptop power adapters/chords or a camera lens) from carry on to personal item in order to make the weight limit for the carry on, and then moved it back. You can also in some cases use your jacket pockets. I've done that more so with airlines that have a 7kg carry on limit (EVA, Jet Airways). As long as the airline doesn't impose a total weight limit for both carry on and PI that strategy works.

Suggest you get a set of luggage scales, and before you leave weigh your bags in certain configurations and plan what to swap from bag to bag. That way you can remain "legal" weight wise and not get stung.

Occasionally.

In my experience, it happens about in 10%-20% of my LH group flights (including Swiss and Austrian).

I have a pretty big roll aboard suitcase which probably triggers more attention than a smaller piece of carry on. In all cases, taking out a jacket or moving something heavy to my backpack did the trick, but I always make sure I'm not horribly overweight. Anything over 8.5 kg can get you flagged.

They might, but you should not worry too much.

If it is weighted and goes over the limit and the check-in clerk does not let you "pass this time" you have other options.

You can swap items form carry-on luggage to checked-in luggage. You can leave the desk and check-in later with some items in your pockets, etc.

The best advice, however, is to check-in as early as possible and use small (not bulky) carry-on luggage (backpacks are better than suitcases).

You didn't mention if you are going to have checked baggage as well, so my reply is in the case where you don't.

I have travelled with Lufthansa in Europe more than 10 times in the past 3 years, and every time I just had a backpack as a carry-on luggage and a laptop bag as a personal item. Because I did my check-in online I just skipped going to the check-in and straight to the security check. Nobody ever weighed my backpack, even though I am sure sometimes it weighed about 10 KG. Also, nobody said anything when boarding or on the plane.