It is clear that we write:
He does cry...
He thinks of her
But, I'm confused whether the second verb should be with 's' or without 's' in this case:
He does cry and think/thinks of her
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Both are possible.
The use of "does ..." is an emphatic. It indicates that the speaker is contradicting the expectations of the listener.
Even though he is a grown man and rarely shows his emotion, he does cry.
So if we treat "cry and think" as one verb phrase, and we are contradicting expectations on both the "crying" and the "thinking", it would be correct to say
Even though he tried to forget his first wife, when the radio plays that song, he does think of her and cry.
(It is easier to write an example in which the thinking and crying are swapped)
On the other hand, you might want to create two clauses, meaning "He does cry, and (then) he thinks of her":
The song comes on the radio. Now, at last, he does think of her, and cries.
Putting a comma helps this reading, but isn't absolutely necessary.