I started with a pick very early on but switched to mostly fingerstyle within a year. I find Fenders the easiest to finger pick, the spacing just seems right to me. I find the T-Bird the easiest to play with a pick, I find the T-bird's strings too close together to finger pick well although I've gotten used to it.
I rest my thumb on the E string except for of course when I play the E, then I pull my hand up a little bit and put it right back on the E string. I do this unconsciously, I had to get up and pick up a bass to see how I do it just then. The reason i do this compared to resting it on say a pickup edge or bridge is because I move my hand left and right depending on how it feels and what kind of sound I want. Closer to the bridge, more percussive, clearer staccato sound, closer to the neck more bassy and blooming sound. More funky towards the bridge and more bluesy towards the neck. I have more control near the bridge as the strings feel tighter.
A technique I've been using a lot especially for blues is using my thumb while palm muting the bass, you can get a very punchy deep clear almost upright like tone like that. This sounds equally as good on the Fender or the T-Bird, it's difficult to palm mute Rics.