Continuing with my troubleshooting. My main problems now are no task-aware debugging with the LPC-Link2, and I still can't launch debug configurations directly.
'Search Project' from the debug configuration has been weirdly inconsistent. It'll simply not find binaries. I can browse to them and they're in the right place but it won't find them otherwise. I just rebuilt all of the configurations and now all three are showing up in the list again.
Of my P&E configs, now two are giving me an active debug button. And now that I switch back to the NXP configs, all three have debug buttons. On P&E #3, copied the application location from NXP #3 but still no debug button. 'Search Project', selected the binary, no change. Clicked back over to the NXP #3 config for reference, then back to P&E #3 - and now the debug button is active! I changed nothing, just switched views.
Launched from the debug config screen and it worked normally. Still no debug button in quickstart. All configs still have an active debug button. Selected NXP #3 and 'debug', got a new error:

Never seen this one before. Target discovery is now stuck, won't cancel. Cleanup doesn't help. Closing the IDE and restarting.
Still no quickstart debug button. All six debug configs are still active. Launching with NXP #3. Now I get the probe selection and I can choose All-Stop. Suspend works, but still no task list in the debug view. Incidentally, reset handling has reset to the default blank setting but I'm not getting a wire ACK error.
Clicked on the flash GUI tool to check something and got this:

Cancelled, made sure redlink server was down, tried again. Got the probe discovery screen this time. Ran the mass erase and it worked. I was trying to recreate something I meant to ask about previously - every time I'd try to run a mass erase, it'd pop up with the probe discovery screen again, but this time showing only one option. I had that happen at least half a dozen times, and now it's not doing it.
Clicked on the blue bug. Still getting an error:

Following Erich's instructions, I've set the debug level to 4 in the debug config. FreeRTOSDebugConfig says it's missing. My FreeRTOS settings don't match Erich's; changed them and rebuilt. Ok, now we're getting a task list. It's slow to come up compared to the Cyclone, but it's there. Back to the Cyclone, still looks good there.
Clicked on the MCUX drop-down from the quickstart debug panel and got another error about no matching launch configs (this time with something other than using: none but I didn't catch it), but now the probe discovery comes up. Launches OK. Tried the drop-down again, captured the error this time:

Switched build configurations. Tried all three, none show a debug button in quickstart. Green button works, but oops.. launched an inactive configuration, not the current one. Remembered I had WFI in idle disabled, re-enabled it. Launched from debug config screen on NXP #1 config, got the probe screen, changed to all-stop.
Restarted the IDE. Now the blue bug is disabled again, and the one in the quickstart panel is still inactive. But if I click down to the quickstart panel and back up to a project file, NOW the blue bug is active. Still gives me a 'disabled' error.
Back into debug config to set the log level back to 2. Noticed semihosting on, which I don't want. Turned it off - and now the debug button is disabled again!

Switching back to the main tab, it now says the program does not exist:

Not in the program selection:

The matching config in the P&E group is also disabled. Exited, rebuilt, no change. Cleaned and rebuilt, still no debug button. Pasted in the location of binary #1 into the location for NXP #3, 'Apply', no change. Checked P&E #3, back to NXP #3, now the debug button is back. Changed it back to the right binary, 'Apply', debug's disabled.
Clicked 'Browse', selected ELF binary. 'Apply'. No debug. Exit, rebuild - nothing to be done. Search project, still no binary found for config #3. Clean and rebuild, still no debug button. Note that it doesn't say it's missing the binary. Still nothing in 'search project'. P&E config #3 has a location of "T4-Debug/T4.elf". It does not show an error - but no debug button. Pasted that into NXP config #3 and I immediately get "program does not exist."
Restart IDE, clean, rebuild, no change. Changed the debug level and semihosting settings back just for the heck of it, no change.
Entering garbage for the application in P&E config #3 does not give a 'not found' error, so maybe it doesn't check?
Switched to build configuration #1 and rebuilt, and NOW all three binaries show up in "search project" again. Select the binary - no debug button. Look at another config, then back, now it's active.
To confirm that the build configurations are correct: Build artifacts are "ADS-SR2" for #1, "MSX-100" for #2, "T4" for #3. This is correct. It was T4-Debug/T4.elf that wasn't showing up in the list. Rebuilding Debug/ADS-SR2.elf made T4-Debug/T4.elf show up in the list. Still no quickstart blue bug.
Trying to replicate the debug button getting disabled. Active build config is #1. Set semihosting off, apply - debug is disabled for an instant but comes back. Switched to build config #3. Noticed that the debugger tab is in a different place in each debug config, and it moves at random when I switch between configs. My Android developer happens to be passing through at this moment and says "that's because it's Eclipse."
Set semihosting back on, loglevel to 4, debug launches normally. Set everything back, no change. Tried the blue bug again - same error. Switch to P&E config #3 and debug, works normally. Clean project so now binary #3 is in fact missing. T4.elf still shows up in "search project", though a file system search shows it's not really there. There's no warning about a missing application. Click debug, project builds and launches normally.
I think that's about all I have in me for tonight. To recap: I'm now able to debug with the LPC-Link2. The reset handling option is NOT set. In fact, I just changed it to 'default' and it works the same; I have no idea what actually made the difference. Disabling non-stop debugging and setting the FreeRTOS debug config properly got the suspend function working. The quickstart menu still doesn't give me a debug option, and the blue bug on the toolbar throws an error. The debug button from the debug config seems to stop working randomly and starts working again when a different config is rebuilt.
I hope that's enough to go on, and to illustrate the inconsistent behavior I'm seeing. I was going to edit this down but seeing as I haven't come to a useful conclusion on some of these issues I'm going to leave my whole rain dance of a troubleshooting session intact for reference.
Scott