1. Everyone of them rose from their seats.
In English, we would not normally use everyone as a single word and follow it by "of them". The "of them" is already implicit in "everyone".
When used as a single word "everyone" is a 'mass noun' referring collectively to the entire group of people. As you are already referring to the group of people as a single entity, the "of them" is inappropriate because it is referring to them as a group of individuals So you would normally just say:
Everyone rose from their seats.
You could say, for example:
Everyone [who was] still sitting rose from their seats.
Everyone in the stalls rose from their seats.
In such cases "everyone" is restricted to a sub-group, but that is not relevant in your examples.
But none of the sentences above emphasises the unity of the audience in all rising from their seats.
2. All of them rose from their seats.
That is a perfectly acceptable sentence, and (in my view) is slightly stronger that #1 in emphasising the unity of the audience in all rising from their seats.
I would not suggest replacing it with
They rose from their seats.
because that is just a plain statement with no emphasis on the unity of the audience and hence does not have the same implicit meaning. But you could say:
They all rose from their seats.
which would be equivalent to your sentence #2.
3. Every one of them rose from their seats.
In contrast to sentence #1, this is acceptable, with "Every one" written as two words. In this sentence, the "one" belongs with "one of them", whereas in sentence #1 the "one" belongs with "everyone". Thus this form means "Every individual person rose from their seats".
This sentence #3 strongly emphasises the unity of the audience and is (in my view) the 'strongest' of the three sentences in that respect.