1) Show the link ID as an image. Look at Factorissimo for what I'm picturing.

2) A button (and perhaps a circuit) saying to cycle it now even though it's not full.

1. I assume you mean an alt-icon. This is possible but it would make it very different from other buildings in Factorio, and I'm not sure if I like that.
2. There already is. You can click any of the buttons under "Destination" or you can send it a planet signal via the circuit network.

Yeah, an alt-icon. Containers in the game normally have an alt-icon, this is fundamentally a container that does something special.

Huh? You can select whether it will teleport to a non-empty target, but then there's a list of planets which you can't interact with, then a line that says "Teleport when full..." and a picker for the planet, which selects a planet, then a check box for "Read working" and a box for "Control signal" that's empty. I understood that to be for sending out a signal when it's activated. It's formatted like an output, I take it it really is an input?

Is something perhaps not showing up on my screen like it should?

Strange. This time it gave me a clickable button for the planet, it let me send an incomplete load. But the circuit I hooked up clearly was reading an output.

If the buttons weren't clickable, it's because something wasn't lining up: the Link IDs were not the same, or the conditions were not met. For example, if you have told the other end to only allow a teleport when full, the button won't be clickable.

The "Read working" signal is an output, which is output when the teleporter is in the process of teleporting. As noted in the mod's description, if you want to activate the teleporter via a circuit network input, you can send it a planet signal.