Wemos D1 mini pins on HIGH on startup

a) you can, but it goes over the RTC peripheral so it is slower
b) the right way is to control the relays with inverted logic. some relay modules close the relay at LOW signal. does your relay module have inverted logic?
c) pins D3 and D4 have external on-board pull-up resistors. all io except of D0 have the internal pull-up option