"Obviously your server can not handle Moodle, a) either because it is too weak b) or it is misconfigured."
OP said on certain processes/usage server would 'mis-behave' ... even listed which/what. So installation is semi working. In the past, had investigated something similar and found that in some .php files Moodle was attempting to get 128M more memory before executing ... that was programmed into the file by a variable. Strange things happen when max memory is reached or exeeded or SWAP space is used.
Anyhoo, OP has a 'developer' looking into it ... let's hope 'developer' can 'weave some magic' for the OP.
'spirit of sharing', Ken