Firstly from my understanding looking at NASA website the ISS cycles the earth 26 times in one 24 hour period.
Not quite that fast. Objects in low earth orbit (including the ISS) can complete an orbit in around 90 minutes.
Time on the ISS is also set as GMT +0. As this is the case, how did the time on earth and the ISS stay synced or is that way it goes that many times around earth in 24 hours?
Think of that as the "timezone" of the ISS. When it's noon in GMT +0 on earth, the clocks on the ISS are set to noon as well. Unfortunately since they're moving around the planet so quickly, the position of the sun and their clocks are not synchronized. Instead, everyone uses the clocks, and just shades the windows to keep from confusing their body clocks too much.
The clocks on the ISS and clocks on the ground tick at (almost exactly) the same speed. So it's no different from being in a cave and setting your clock by a radio station thousands of kilometers away. It's not the same clock as other people nearby use, but it doesn't make much difference to you in the cave.
...theoretically if the earth stopped spinning there would be no time.
The earth's rotation can be considered a clock. But there exist untold others. Galaxies, pendulums, chemical reactions, and molecules all exist and can interact with their environment in ways that display periodic behavior over time. Deep in your cave, you may be unaware of the earth's rotation. But your cells still run biochemical reactions that make you alert and tired periodically.