You're posting a lot of separate problems but I suspect they are all related.
You're not showing us enough of that screen to make an informed decision.
Why are you using such an old version of Moodle? Am I right in thinking that it has a lot of local customisations?
Yes, you are probably right they are connected.
1) THis moodle version is 2.5 from our client's legacy, after we are done with this migration we will be upgrading it. Now I wonder if php 5.6 is the one giving problem for the moodle 2.5
2) I tried increasing memory limit & timeout on php yesterday, and I thought it helped the error 500, however this time, i can click around longer before it comes up again!
I don't remember for certain but I'd be surprised if Moodle 2.5 didn't work (more or less) ok on PHP 5.6.
You HAVE to find those logs for the 500 error. Without the underlying problem being found you'll never fix it.
