There is a list of Apple Hardware Test versions at …
https://github.com/upekkha/AppleHardwareTest
You will need to figure out the correct one via the commands listed there, then you can download the .dmg & burn that to a CD. If you burn it correctly it will be bootable (silly me - I can't recall if you burn the 'mounted disk image volume', or select the image itself to burn - try both in Disk Utility).
TTD does different things - AHT 'knows' what the hardware should be, so it can perform specific tests, TTD is generic.
I got burned with Tech Tool Pro & have been wary of it since - it deleted essential filesystem features via it's repair options, Disk Warrior fixed it making the Mac bootable again.
I'd suggest you make a new issue to detail the kernel panics - someone may be able to help you find logs & get a better idea of what causes them, diagnostics may not see all issues, but it is a good idea to run them.
Rember is another option for checking RAM…
http://www.kelleycomputing.net/rember/
Start from cold if you can, some issues can be temperature dependant, re-run to get the Mac up to operating temp.