How is the Apple II text flash mode timed?
It's a fixed timing signal generated by an NE555 timer (at position B3 on the original II+).
(From the Apple II Reference Manual / Red Book p.151)
Entering the values of C/R1/R2 into an online calculator for NE555s will give about 2.1 secondsHz cycle time (*1)
The II's video logic used bit 2^6 and 2^7 to control the level in text mode.
7 6
0 0 - Inverted
0 1 - Inverted depending on the timer
1 0 - Not inverted
This is as well visible in the (screen) character set:
This as well shows that the Apple II character generator only holds 64 different glyphs - using above logic to produce them in Normal/Inverted/Flash.
There is no hardware cursor. It's essentially done by setting the two top bits of the character were the cursor should be shown to 01 (*2).
For the second part:
an extension would be why is the cursor is affected but not the text display?
I'm not entirely clear what you describe/ask.
*1 - These are analogue parts with up to 10% or more variation, so observed speed may vary.
*2 - this is also, why pressing Reset may result having the 'old' cursor still blinking while a new one is displayed.