Update: I decided to go into the code again and Intellisense this time showed me an error in the component about an undefined GetOwner () and as a result the component did not compile at all (earlier the error did not appear and the project was successfully packed). And apparently due to the fact that I switched the variable, and TickCharacterPose remained - there was an animation overlay, albeit a very strange one. After I include Character.h and recompiled - everything seems to work as it should (I really hope so).
I solved the second problem by simply increasing the delay between dropping the weapon and turning on its physics. Apparently due to the fact that the delay was equal to 0.1, weak computers did not have time to process this task. Actually, this is not related to the topic, but perhaps it will be useful to someone.
It is very difficult to find detailed information on network problems. For example, every developer who makes games through the listen server will be affected by this fix. I really wanted the flexibility to configure this initially, as I read in one of the topics (written by the way many years ago), it would be a great idea to make a boolean switch for these modes somewhere right in the engine.
That is why I tried to describe everything here in detail, in order to at least slightly expand the view of all these problems for future seekers of answers.
By the way, if you work with the Advanced steam session plugin, I had a problem with finding matches that occurred if I packed the game into an archive and the unzipped copy did not connect to the one that was archived before. So be careful with this. But nowhere did I see even close advice about this. (very weird btw)