The washing machine drain is connected directly to the main drain pipe without a stand pipe or trap. Calling this out as improper connection. The sump pump (discharging the laundry sink water), is connected the same way. Reporting this connection as improper, the sink should discharge by an enclosed pump, not an open sump well. Opinion on the condensate drain, will the trap hold enough water to maintain a seal?
No the condensate drain will dry out when the HVAC sytem is off.