"Would space-time get disturbed like the wake and bow wave from a boat [...] or oscillate like the air around an implosion if said planet/star was made to disappear?"
Space-time is a concept introduced by Einstein, and is central to his hugely successful theory of general relativity (GR). AFAIK, the concept does not exist outside GR, or possible extensions to it.
Now in GR, mass-energy cannot be made to disappear; in fact, GR makes no sense at all if mass-energy is not conserved.
So your question is equivalent to asking what happens to something which exists only in a theory, assuming the theory is hopelessly (internally) inconsistent.
And the answer to such a question is ... that the question is meaningless (and so you can make up any answer you like).
Of course, you could change the question somewhat, and add that in disappearing the planet/star conserved mass-energy! If you did that, then the answer to your question would be "It depends"; specifically, it depends on the details of how the disappearing is accomplished (and it is not possible to answer the question without those details).
Alternatively, you could ask what would happen in some theory - not yet developed - that, like GR, incorporates the concept of space-time, but permits planet/star disappearances (of the non-conservation of mass-energy kind). And I'm sure you know the answer to that! (HINT: you need to say what that alternative theory is, before you could answer!)
Speculation can be quite fun; however, here in the Q&A section of BAUT, there is speculation that cannot be answered (check out the BAUT rules if you don't know why).