Sling - cool ! Not embarassing ...we all overlook simple faults sometimes, and we all have to learn.
J121 - yep thats almost certainly yr culprit. Is the header pin itself melted/burned ? Or just the wires connector ? If the header pin is then ideally the whole header pin block should be replaced, you may get away with resoldering just that header pin - wont look pretty and may only be a temporary fix. If the actual connector block is showing signs of overheating then really that should be replaced also.
The clock - sounds like testing fine, so could be intermittant fault. In game during certain modes such as Clock Chaos , the hands of the clock will turn to show amount of time left in mode, points etc. Importantly it should be at Midnight at game start and always return to Midnight after any mode ends.
TZ is notorious for issues - there are just so many toys/gadgets that gremlins can get to work on ....the gumball machine, the clock and the powerball switches/sensors being the most problematic. Don't let it put you off though, all these issues are really well documented on the web and are easily fixable.
It can also benefit from lots of tweaks to help the gameplay , eg the Rocket kicker should always fire the ball straight across the playfield into the Hitchhiker lane and on to the pop bumpers, the lock lane sometimes needs adjusting to prevent bounceouts and lots of others. Might be worth getting someone who knows TZ *really* well to have a look at yr machine and assess how it plays, and how it is setup. As a new owner and with such a complex game there are probably many aspects, features that you may not even be aware of yet.