Fascinating set! At first I was a little surprised to see those 3 choices grouped together, but then it dawned on me, "But of course they belong together, they are all shape-shifters!"
Base abilities seem good; Alter Self can be potent but it's limited to once per day.
Double: This preserves a very changeling, doppleganger feel to the race while keeping in with the A5E conventions. I think this is a solid entry.
Hamadryad: The only ability here that gave me pause was the Vines power. Since it appears to be at-will and usable as a bonus action, I can see it being problematic in the hands of devious players; for example, as a bonus action I could keep making Grapple checks every round in addition to my normal action; even if the strength of the grapple is low, it could still be very annoying and eventually the dice gods will win in my favor.
A Proposal: What if it normally can work like a Mage Hand, but the hamadryad can add a temporary boost to grapple, pull, or push or manipulate a larger object? This boost would be usable once per short or long rest.
Everything else with Hamadryad looks good!
Zoanth: Here I feel they could use a little buff. The 1d6 natural weapon damage is nice, but not as useful if you have access to higher-damaging weapons. I might make the natural attack a finesse attack as well, so a Rogue Zoanth might be able take advantage of the ability.
A Couple Proposals: While Shapechanging, perhaps the Zoanth should also gain Darkvision and/or +5 movement? And taking inspiration from Eberron's "Shifter" race, maybe a "Once per Long Rest" ability could give them some temporary hit points?
Thanks again for inviting feedback!