I can answer both as I've had personal experience with similar scenarios but not only with CX...

1. LHR - If all connections are on the same ticket , then YES, you are allowed to thru check via LHR on an overnight, ie next day - but it must be first flight out to your final destination. It cannot be the 'usual' 23 hour rule. But if you're on seperate tickets, even with connecting at LHR with first flight out, it is then not permitted. I had a CX flight from HKG-x/LHR-AMS and CX will not thru check since it was an overnight on a seperate ticket. The agent explained to me the reasons why and it was on her computer specifying the fact. When I flew SQ SIN-x/LHR-CPH, the agent could thru check me and the connection was the first flight eg 6am flight out. The agent explained that these rules are specifically for London Heathrow.

2. T4. I suspect that DL is the ground agent for MH and they cannot thru check on seperate tickets. Generally MH is ok to thru check on difference alliances even, so connecting to CX would have been fine. There are interline arrangements on - but if the ground agent isn't well versed in issues like this - maybe the DL agent forgot that MH has joined OW !

US / Canadian airports will also not allow connecting bags to stay overnight. eg security issues.