Your mention of the term “snap election” indicates a misunderstanding about how the U.S. House works. In short, no, there would not be a way for the entire House to have to stand for reelection as happens in most parliamentary systems. Members of the House are elected to serve two year terms. If they leave their post for any reason there is a special election held to fill the seat. Other than for them resigning or being removed (rare) they serve their two year term. My understanding is rules for how special elections are held vary from state to state. There is the possibility the seat remains vacant for a period of time.
The majority party is the party that has the majority of members at that given moment. In the new Congress the Ds have a 10 seat majority in the House. I think that’s the number. Anyway, if a sitting President were dumb enough to appoint 6 members of the House to positions in his administration under this circumstance then yes, his party would lose the majority in the House and the opposing party would be able to vote in a different Speaker, at least until the special election to fill those seats were held. I can’t imagine a scenario where that would happen.