Just got SE2 recently and have tried it a few times in my mainly steel tube and fabric aeroplane.
Initial tests have indicated that it's ADS-B out transmissions are not picked by OGN-R ground stations as well as those of either Pilot Aware or PowerFLARM but still experimenting with placement in the cockpit. Nevertheless, I think it would be better if it had an external antenna connection (to connect an antenna external to the set, not necessarily to the aeroplane).
In any event, the ADS-B out might be being picked up fine by other aircraft in the air!
I will leave others more expert to answer you other queries in detail but a few brief points:
Presumably your transponder does not have extended squitter, otherwise you could achieve ADS-B out in another way. In any event, SE2 has ADS-B in too which can be displayed on Skydemon etc.
I understand that ADS-B out transmissions cannot be picked up by all UK ATC units and, in any event, only certain SIL levels can be used by ATC.
SE2 does not transmit a squawk code- that it still the job of the transponder- although SE2 can/should be programmed with the aircraft's unique ICAO address (found on the G-INFO page for your aircraft).
You can now use a transponder and SE2 at the same time. However, the manual states:
5.1.1 United Kingdom
EC Devices do not replace the need for a transponder when required by airspace/flight rules. Aeronautical Information Circular (AIC) Y 141/2019 allows for the use of EC devices simultaneously with ATCRBS or Mode S transponders which are not Extended Squitter (ES) or ADS-B OUT enabled. If your aircraft is ADS-B OUT enabled, you must disable the transmit function of SkyEcho from the configuration settings.