Task Scheduler & CEIP

If I completely disable Customer Experience Improvement Program, will Task Scheduler be fully functional?

Yes, CEIP is in no way a dependency for anything else in Windows. The relationship is actually, the other way around.

Tas Scheduler is a native Windows function meant to run programs automatically based on some conditions.

CEIP is a piece of spyware that attemps to submit your private data and usage patterns to Microsoft. Part of its functioning is relies on Task Scheduler (you can see how many default tasks Windows bundles related to it). Disabling CEIP only means that the related tasks won't run anymore, or running them wouldn't (or shouldn't) send data out there, but TS will remain unaffected.
The opposite is true. Disabling TS will break CEIP as a side-effect, as the data-exfiltration tasks wouldn't run.