Can you take a photo of the eachine e010 board? I have one that I purchased last month and it binds just fine.
I have a sport cub too (the big one yes?) you have to assign a switch to channel 5, then safe will work, there should be no mixing in the remote, but the safe itself does some mixing when in novice mode.
The lemon rx, yes of course you always have to select the external module when binding, there is no internal "chip" on the t12, everything goes under MULTI under External RF in the menu, I have attached a copy of a screenshot of mine that is bound to a LemonRX stabiliser+, I don't have to rebind it each time.
As for rx, unless I'm using the lemon, my go to rx is the Frsky XM+ (as most of my models either have an FC or are drones).