Electrical – I lost power to half the house

Since neither the washer nor the dryer breaker are tripped, yet the main did trip and power to those circuits cannot be restored by resetting the main, I suspect either the circuit breakers, busbar section, or both have become damaged inside the panel -- this means the next steps involve turning the main breaker off and disassembling the branch circuit panel, looking for damage to the busbar tabs and breaker contact clips, as well as the breakers themselves.

If you're not comfortable doing this yourself, now's the time to get an electrician in; while you have them in, have them replace the misfit breaker with one of the correct type -- i.e. a THQL120 instead of the BR120 that was improperly fitted there, and install a blank over the gaping hole in the main panel.

(It's likely that the misfit breaker was making poor contact in the first place and just quit making contact with the busbar during whatever caused the main to trip. Replace it with a THQL120 and things should be good.)

Don't the two hot legs need to be balanced if it's line to line to compensate for the fact that there's no neutral?

No, if there is no neutral then all the loads must be connected across phase to phase. In US terms this would mean that there are no 120V loads that need to be balanced.

I'm not convinced you have two hot legs from a US style split-phase supply.

The Chinese standards seem to be pick and mix of other countries standards. Note that Red=Live, Blue=Neutral, Green/Yellow=Earth is a valid colour combination in Australia.

The four "two-pole earth leakage breakers" (tengen DZ47LE-32 C16) are marked with N for Neutral at one connection.

I would therefore, naively perhaps, expect the blue wires to be neutral and to have zero or very low voltage with respect to the green/yellow ground wires.

I don't see any normal bus connectors, it seems the installer used red wires to construct buses live and instead of the neutral terminal that would be found in a UK consumer unit.

CE marked C16 circuit breakers should be 16A breakers with a "normal" sensitivity range.

The blackening of the top screws of the old breaker is also concerning. Is too much resistance on the wire or screws/breaker a possible cause?

The clamping screws were not done up tight, the cables were not inserted correctly or the switch was passing too much current. Or all three.

Example, for comparison purposes, of an old and unusually neat UK equivalent showing copper bus bars at bottom connecting live feed to single-pole breakers, terminals at top for neutral and ground (these terminals are sometimes also called bus-bars). Note the copper bus bars normally have a plastic shield, and normally there is a cover that shields all the wiring with holes for access to the breakers only.

- Image by RF Lighting

Since China is not the UK, there may be entirely different exemplar Chinese installations that bear little relation to this. The question shows an installation where every breaker and switch has two poles. This means there is no need for a neutral terminal connector strip. However the neutral wires linking adjacent "earth-leakage breakers" form an improvised bus.