No you don't have to carry the ERSA. You only have to carry documentation that allows you to SAFELY conduct your flight.
Rule of thumb 1) use common sense and 99% of the time you wil be OK
Rule number 2) establish a good relationship with you training organisation and stay in touch after you get your licence, ask them questions, not pprune.
Rule number. 3) buy a set of reg's
Rule number 4) well I forgot what rule 4 was but it must have been important.
Rule number. 5) don't drink and crap on on pprune.