Opposite of implanted would be removed. Eradicate is a little too strong it means totally destroyed. Install is not really used for people, until we become half cyberhumans ; )
So sentences 3 & 4 eradicate.
Sentence 1 is fine and the most common form. But you could use …are just breaking through (the gums).
I’m not comfortable with this sentence since I can’t understand what exactly he wants to have done. As I see things, an implant generally isn’t visible and an implant is often something artificial. Technically, the only part of a tooth that can be implanted is the root. So, if his original tooth (along with the root) had been removed and is now to be put back in, then I might say “re-implanted”.
A typical verb to use when you talk about removing a tooth is “extract”. You can also say “pull a tooth”.
As Stew mentioned, you don’t “install” teeth (unless you’re trying to be humorous ;)).
The child’s teeth are coming in.
or
The child is teething. (This is the verb teethe.)
He wants his front tooth implanted, or reimplanted, as the case may be. We might say he wants an implant to replace his front tooth.
The opposite would be to say formally that he is having his front tooth extracted, or in ordinary language we would say he’s having his front tooth pulled.
You can also have a tooth capped, rebuilt, reconstructed, crowned and some other things, depending on what has to be done. You never have a tooth installed.