As usual with riding in London, confidence in traffic is your friend. By and large, drivers expect there to be cyclists around and while they won't visibly give ground, they're used to not running you over. Stay fast and stay out from the kerb, and you'll be fine.
Lea Bridge Road is fine, as is the road up through Woodford and Buckhurst Hill. The A road through Epping Forest can be a bit unpleasant with boy racers, but you turn off before that bit.
As for the tube, off peak (ie not Mon-Fri rush hour) you can take full-size bikes on the Central Line from Leytonstone if you want. There's a chance you'd get them on at Stratford, but it's against the roolz ...