Of course you should try resetting the camera to defaults if not already done. Lots of stuff going on there, broken battery door, flash issue.
Sorry, but the only people I know who have problems like these are unlucky ebay purchasers, who unfortunately bought someone's headache or problem.
If you can't put a battery of known quality into the camera and turn it on reliably, then what is it really worth. I guess a great daylight only camera. Safe to rely on until it gets dark? Sorry, thats what I see here.
Hope you can figure it out.
Stop using the built-in flash. The built-in flash is not strong, anyway. If you want to do a lot flash photography, then buy an external Speedlite, and move on. Stop using the built-in flash. Only use genuine Canon batteries, too.
It sounds like it came with a lot of batteries probably all of which are seriously compromised. The peak current demand from that pop up flash is quite high and it probably pulls the voltage too low when it tries to activate. The flash could be the problem (i.e. defective component in the inverter or storage cap that it feeds) but I would try it with a known good battery first.
That indicated charge level on your third party charger doesn't tell you anything about how the battery will actually perform when called on to source a heavy current. Decreased capacity and increased internal resistance with aging/abuse result in a very large voltage drop under load even though the battery indicates a normal or near normal cell voltage/charge level when not under load and this problem is also greater with many third party batteries which, even when new, can't deliver the same current/voltage profile under heavy load as a Canon OEM battery. There are two reasons Canon OEM batteries are more expensive, one is that you do pay for the Canon name BUT you also pay for the basic design and the build quality that lets the battery pack produce heavy current without excessive voltage drop in a real world environment.
The AH rating of a battery pack is specified and measured over a long discharge period and doesn't actually represent the total current the battery can produce for 1 hour. Better quality packs come close to the standard of producing the rated current for 1 hour representing a very heavy discharge condition while lesser packs will last far less than an hour under this discharge profile. In the real world of digital cameras, battery discharge involves a lot of high peak current events and batteries in poor condition or those simply not well made from the start won't fair well when called upon to deliver high current because of their effective higher internal resistance.