每个人都...... Why the 都 is compulsory in the sentence.
都 is an adverb for 'All' ; 'without exception'

"每个人带着甜蜜的笑容。" is not grammatically wrong, However, having the adverb 都 before the verb 带着 (carry), add emphasis to the fact-  'All of them/ without exception' 


'人人带着愉快的笑容' = 'everyone carry pleasant smile on their faces'
'人人(都)带着愉快的笑容' = 'everyone, (without exception), carry pleasant smile on their faces'


It would be easier to see the difference if you replace 每个人(each person) or 人人 (everyone) with a term that hasn't already indicated the meaning of 'all'

For example:

'外面的人们带着愉快的笑容' = 'the people outside, carry pleasant smile' (we don't know it is some of them or all of them)

'外面的人们都带着愉快的笑容 = the people outside, all carry pleasant smile' (now we know all of them are smiling)

Side note:

甜蜜的笑容 = 'sweet smile' , more suitable for describing smile of beautiful or cute girl