Figured it out. It's a graphics hardware issue and I just needed to try a different renderer in settings. I feel stupid, but it never occurred to me that I should change it from "Automatic" to something else. Direct3D 9 and GDI render modes both work fine for me on both games.
I might not have figured it if I hadn't found Nyergud's crash research thread, which made me realize the culprit was cnc-ddraw. Thanks to FunkyFr3sh's reply on this post, I learned cnc-ddraw requires OpenGL 2.0 and ts-ddraw requires OpenGL 1.2. My Linux laptop is old and has an Intel 950 GMA chipset running OpenGL 1.2, which explains why TS ran and TD/RA didn't.
Thanks for the quick reply, FunkyFr3sh! I was actually trying to install the full game and not just the online play, although it looks like the CnCNet app already got installed for me, possibly when I installed Tiberium Dawn or Red Alert? The PortingKit version is what's linked to at https://cncnet.org/tiberian-sun.
Anyway, I ended up opening the Wine wrapper and installed the EA Freeware version into it, then applied the CnCNet patch. Seems like everything's working now.
Instructions below for anyone who needs them:
Attempt to install C&C Tiberian Sun + Firestorm from Porting Kit.
It fails at the error I originally posted about, but should install the necessary Wine wrapper, winetricks, etc.
Download the EA freeware version OfficialCnCTiberianSun.rar (which contains the C&C: The First Decade version).
Right-Click on the C&C Tiberian Sun + FireStorm app installed by PortingKit,
and select "Show Package Contents"
Extract OfficialCnCTiberianSun.rar into C:/Program Files/
so that all the game files end up in C:/Program Files/EA Games/Command & Conquer The First Decade/Command & Conquer(tm) Tiberian Sun(tm)/SUN/
The above path gets used in the registry file in step 9 below.
Delete the C:/Westwood directory created by the PortingKit installer, as it's no longer needed
Back at the root of the C&C Tiberian Sun + FireStorm app package, open the Wineskin app.
Select Wineskin "Advanced"
In Wineskin Advanced, select Tools -> regedit to open Registry Editor
In Registry Editor, select Registry -> Import Registry File.
Then, select C:/Program Files/EA Games/Command & Conquer The First Decade/tibsun_regadd.reg
Then, close Registry Editor
Download the "Patch for existing installations (Windows)" (TSpatch.exe) from https://cncnet.org/tiberian-sun
In Wineskin Advanced, select "Install Software", and select TSpatch.exe
The patch will automatically open the C&C launcher, close it to finish work on the Wine wrapper.
In Wineskin Advanced, in the "Windows EXE" field, select TSLauncher.exe, the path should look like this: /Program Files/EA Games/Command & Conquer The First Decade/Command & Conquer(tm) Tiberian Sun(tm)/SUN/TSLauncher.exe
In Wineskin Advanced, select "Test Run" to make sure everything works.
Close Wineskin Advanced and launch C&C Tiberian Sun + Firestorm.app
