I play FH2 on Linux (Ubuntu). It runs very well. The only problem I'm aware of is that I can't see FH2 servers in the server browser. However, if I join them by manually entering their IPs, they will show up on the Favorites tab, where I can favorite them. Then, I don't have to keep entering IPs to manually join FH2 servers. I don't know about 2160p resolution. My monitor only does 1920x1080, and that works fine in BF2/FH2, for me.

I believe the reason I can't see FH2 servers is because the master server my BF2 install is using doesn't have FH2 servers on it. I read somewhere that FH2 uses its own master server. I believe the reason my game isn't using the FH2 master server is because I start the game using BF2Hub. The reason I do that is because that's the best way I have found so far to start the game and get into a game of FH2. I think the FH2 Launcher and its server browser and Update / Repair functions are what get you the FH2 master server. However, the FH2 Launcher crashes when using a lot of its functionality on my Ubuntu/WINE setup. For me, these things trigger a crash (I can provide more details to anyone interested):
-Hitting the PLAY button. The game will attempt to start, but will remain a black screen and eventually crash, with a winedbg error window.
-The SERVERS button generates a long error message in a window and fails to function at all.
-In Update / Repair of the Options menu, Check for Updates offers an update, but gives two error messages when I click Update and fails to do any updating and seems to require being killed with a kill signal. The Repair option has the same results.

The one thing I've found to work with FH2's launcher/updater software is that the Options seem to be settable and the correct results of those option settings appear to show up in game, when I run the game using BF2Hub.

The fact that the game seems to run flawlessly in WINE with BF2Hub and essentially doesn't work at all in WINE with the FH2 Launcher/Updater is peculiar.

If I could have access to the FH2 Launcher and Updater source code, I might could figure out what the problems are with them in WINE, if I were to find the time to dig into it.

The way I installed BF2 in WINE, on Ubuntu, was to install BF2 into WINE's pretend Windows file system (the way you always install Windows software in WINE; in Ubuntu, you can just double-click on installation files). I then followed directions I found online to install and setup BF2Hub so that I could play BF2 online. I then installed FH2. I originally tried the FH2 Launcher that the FH2 people recommend you use, but it didn't work. I'm pretty sure it crashed. So, I had to use the 3 separate installation files to install FH2. All of these files were in the Downloads section of the FH2 website. Those files worked properly and successfully installed FH2. At this point, I can run FH2 by running BF2Hub, selecting the Forgotten Hope 2 mod, and then clicking PLAY BATTLEFIELD 2. I can then join FH2 servers by finding them online, using sites like gametracker.com, and using the option to manually enter an IP address, in BF2. Once I have played on them, they will show up in the Favorites section of the in-game server browser.

I consider this working very well, given the fact that some games won't run at all in WINE, at least on certain computers, if not all computers.