Usage of begins: Everyone looks suitably depressed at this remark and begins to

[Aikuzo]

Everyone in the pub looks suitably depressed at this remark and begins to think of all the price increases that will follow.

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Shouldn't it suppose to be "begin"? So sentence like "Everyone are waiting." is wrong? Also, is "begins" the verb on that sentence?
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I quit studying English years ago and have just come back to study the basics.

[Erik]

Everyone is singular. Think of it as "Every-One".

[Aikuzo]

That makes sense. So it's like "Neither him or her is giving up.". Hey thanks Erik.

Btw, is that period after quotation mark 'giving up."(.)' correct???
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I quit studying English years ago and have just come back to study the basics.