Are there any sources for this, or can anyone make an educated guess? It seems like most of the men who were of fighting age would have been sent to Troy. Would this have opened up the country to banditry? Were supplies, and reinforcements sent to Troy on a regular basis? Did governing change due to the leaders being abroad?
If all of the gods were busy fighting in Troy, would this have any meaningful change regarding worship in Greece? For example: would less important, local gods prayed to more often?
I'd be curious about any other aspects of life in Greece during this time as well.
I think it's important to point out that there is no clear evidence that the Trojan war even happened. How the war affected belief systems or government can't really be answered because we can't say for certain such a war even happened. There are a few FAQs on Troy that explain the evidence:
Did the Trojan War actually happen? by [deleted]
How did soldiers fight in the time of the Trojan War, specifically in the events detailed in the Iliad?by [deleted]
This second one explains the relevance/problems with using the Iliad, which you seem to be referencing in your gods question.
The Trojan War would have happened in the late Bronze Age, so you can look at archaeological evidence for what life was like in that period. This AMA with a Bronze Age archaeologist has a lot of interesting info on the period and can point you in the direction of further info: