Do Puppeteers get credited with the cast if they are not listed in the cast section?  Also, if a company that does the puppeteering is credited and then individual puppeteers get credits, would the company receive a group divider like in the crew?

My gut says no, at least on the second question.

Quoting primetime21:

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Do Puppeteers get credited with the cast if they are not listed in the cast section?

Yes, they should be included no matter where their credits appear.

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Also, if a company that does the puppeteering is credited and then individual puppeteers get credits, would the company receive a group divider like in the crew?

For me, that would depend on how the credits format this, exactly - for me it's a matter of "I'll know it when I see it". But generically speaking, company headers are certainly not limited to visual effects. The contribution rules simply state: "If a company name heads a group of crew, use the Group divider to enter the company name." That isn't limited to any section in particular, there's no reason that "if a company name heads a group of crew" couldn't be applied to a puppeteering company if there are valid credits listed under it.

Quoting scotthm:

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No reason except that puppeteers are "cast" and not "crew".

Good point.