Even with a damaged socket it could start up fine sometimes. It depends on the extent of the damage. Same thing with the CPUs...they could work and then not work or just have one bad core and still work otherwise. It all depends.
I really wish I could help more but it looks like its pretty much ran its course. Since you were sure the bios was updated prior to trying the new CPU, it should have worked. Everything was installed correctly and double checked but still the issue.
You said the new CPU worked before but now having issues...I just really wish that there was another system to try the CPU in to know for sure. If you got the new CPU through eBay I would suggest a return if that is a possibility.
Honestly I had a Dell laptop my wife got for me when I was deployed and it ended up having non stop problems due to Dell's horrible drivers destroying the battery charging abilities and I never trusted them since then. I wish I could be of some actual help but aside from everything we went over I can't think of anything else that can be tried other than using another T7250. At least it should work fine again, and if it doesn't then it would lead to possibly be the CPU socket. It could even just be a loose wire inside the case causing the issue.