"Do you REALLY think that Trump is going to win the (next) election?" (the next election> next elections) next is optional. No second 'do' and Stress on REALLY if you are doubtful that they think this seriously, or that you personally think this is improbable. "You really DO think that Trump is going to win the next election?" This is that you are hopeful Trump will win the election and are looking for this person to confirm that they also DO think so too.

In your particular example, it doesn't sound right to add the extra do. In your case, to be more emphatic, you would just strongly emphasize 'do' as follows: 'You really DO think that Trump is going to win the next election?' We DO do it in other situations...like in this sentence. 'You DO do that, you know?' 'They DO do that in other countries.’ Sometimes you'll see double words occurring for emphasis; it may happen with ‘Have’ also. ‘Do you HAVE have your homework, or did you leave it at home again.’ This is much more a casual speech thing and isn't grammatically correct. Sometimes ‘that that’ can occur in a sentence, but it's because ‘that’ can serve different grammatical roles. ‘And I hope that that truth may be self-evident.’

The emphatic “do” is not used in question form, because it’s too redundant. You could emphasize “really” if you want confirmation: Do you REALLY think Trump will win? Alternatively, if you’re already more or less certain of the answer, you can use the emphatic “do” with a tag question at the end: You really DO think Trump will win, don’t you?

(American English) - the second DO is optional. but sounds good as is. It is there just to place extra emphasis.

If the first ‘Do’ is a typo for ‘So’ then it’s okay