B737 Captain reported the anti-ice; left engine; and both wing lights malfunctioned during climb-out. Along with communication problems with Dispatch; an additional issue with the filter bypass on the right engine occurred as the flight crew diverted to another airport.

Narrative

While on initial climb-out from ZZZ; we had the engine and wing anti-ice on" - everything was working fine. Then about a minute later; I looked back at our anti-ice and noticed the left engine and both wing lights were off. After reviewing the checklist it instructed to keep the aircraft clear of ice. We continued to climb to cooler temperatures and higher altitude until clear air. After that I sent Dispatch a message to call us and was given a frequency. The issue was trying to communicate with Dispatch and Maintenance Control. The frequency was XXX.X for commercial radio. I requested new frequency and Dispatch said commercial radio is the one that does it. This is the second event this has happened to me where communication with Dispatch is impossible. We had to communicate through ACARS and that takes lots of time and heads-down.We decided to divert to ZZZ1. Then we got a new problem; "filter bypass" on the right engine. I didn't bother to try to communicate with Dispatch again because it was too time consuming. Flying over the USA and having these issues - it's unacceptable."