Do cell phones still have delay, latency or breakup issues these days?

Yup, it can vary from the same phone model, same carrier, or several different models and carrier for a given area depending on the distance from towers, signal strength of the individual antenna on the phone(s).

Yes, and they always will. 

Yes, every update cost storage.

There can be breakups, if you are not close enough to a tower, or your carrier's service is not dominant in your area. Also, if the person you are calling has a sucky phone, there will be issues.