I finally got this to work but will write my experiences here in case someone else runs into the same problems and can possibly get some clues from my trial and error experiment as this was very tedious and time consuming.
I didn't keep my mind open enough. I managed to get the RR-net MK3 to work finally. My c64c, that I, in previous posts, have been writing about saying it doesn't work, does not... yet anyways... The problems is something with that particular computer I have, possibly the expansion port itself. I was however able to use it to flash the RR-net with the "Alternative firmware". But after that I could not get it to respond to Warpcopy (the server wouldn't tranfser), nor did the DHCP work. Only one or two frames got transfered and then it crashed/hung. So, I gave up briefly. BUT then the other day I tried again. I did it with a more systematic approach. Started an old winXP machine up since I thought the network was "noisy", lowered it's ethernet to 10 Mbps, disabled the Wifi adapter, connected a 10 Mbps switch in between etc. It still didn't work!
However, I decided to try this WinXP setup with my two old breadbins C64s. After a while of tinkering I finally managed to get Warpcopy to run on both of them. My one breadbin c64 with a replaced PLA, semi broken SID and one broken CIA worked almost immediately with Warpcopy and could get the Warpcopy server transfered to it. The second breadbin(has all original parts, nothing replaced) at first simply just crashed immediately when switched on and RR-Net MK3 connected. After a few powercycles it started to transfer the Warpcopy server but it got interrupted after a while. I did try this over and over again and more and more of the server got transfered. I had to power cycle in between the tries as the c64 did hang. After several tries it finally got transfered and I could use the Warpcopy program from my WinXP PC ro transfer disk images.
After having success with the two breadbins on the WinXP computer, I decided to retry to connect it to my LAN at home and my Windows10 computer. This also worked for both of them. So, Windows10 and connecting the RR-Net MK3 to a fast speed network also works and also confirms that it is not the RR-Net MK3 that is faulty but the c64c computer I have. The breadbin c64s I have can handle the DHCP and static IP adress assignment, Warpcopy etc in/with RR-Net MK3. I have also managed to run Contiki OS on them with RR-Net MK3.
I also tried again today and the breadbin with all original parts had the same problem with crashing when switching it on, transfering the Warpcopy server took several tries and in general it behaved erratically at first. After a while it started to work so I think it also needs to "warm up" a bit before working with RR-Net MK3. This is most likely depending on that machine and not the RR-Net. This machine has not been switched on since the mid 90s so it might need to be used more frequently and then it might work fine when switching it on with RR-Net.
So, my conclusions on getting RR-Net to work:
1. Have a very open mind about the c64 you connect it to and remember these machines are old. It may seem to work completely if it is possible to load a game or similar on it but some parts really don't work, i.e. the expansion port.
2. It's good to have several c64s to try on. It might not work on one but on the other.
3. Be patient, and try several times for instance with WarpCopy, if an improvement in data transfer is seen.
4. Let the c64 be on for a while before trying to work with the RR-Net MK3.
5. Windows 10 or Wifi/High speed ethernet routers/networks are not really disturbing the RR-Net MK3.
6. Some network knowledge is good to have in case of doing tedious tests to exclude possible errors from a network the RR-Net MK3 is connected to.
The above is tested with the Alternative firmware that I was able to flash on the RR-Net MK3 as I had the correct c64c for correct flashing. I do believe that the original firmware would have acted the same on my machines but this I haven't tested and thus can't confirm. If flashing the Alternative firmware onto RR-Net MK3, make sure the instructions on how to do that is followed correctly!
I will try to get the c64c, that I have with the "ALDI" shortboard, to work later on as well but for now I am quite happy. I believe the expansion port on that machine is involved in some way but have to do further investigation.
Hope this helps someone else in case of problems,