Commas that are sentence flow-stoppers ... Can I leave them out?

The commas have a role, to set off the state name or the year because either of them is clarifying and defining the city or date. In a sense, they are in apposition to the bit that comes before. In "Prague, Czechoslovakia is a great vacation spot," the single comma isolates the subject Prague from the verb is. Your other examples, with the exception of the first one, have similar problems.

Most of your example sentences wouldn't use a comma in the positions you noted. If we replace the complicated phrase with a simple variant:

1. The performance took place on Tuesday at the State Theatre in Ithaca.

2. Tomorrow is Helen's date of birth.

3. Prague is a great vacation spot.

4. Paris will host the Olympics.

5. Dallas is the home of this special event.

6. Syracuse will play host to this year's Rock of Ages music festival.

Examples 2-6 should not have commas. Example 1 could have a comma where you noted:

The performance took place on Tuesday, at the State Theatre in Ithaca.

But this usage is optional and can be removed if you find it interrupting the flow of the sentence.