what would be the best solution for decreasing transaction fees to rational level? Rational being that it could be rational to use these transactions in every day payments.
One solution exists already (Segwit 4MB blocks), but it's not fully adopted yet. It will take time for Segwit to have it's full impact on fee reduction...
Do you have any idea why Segwit adoption has stalled in the last ~3000 blocks?
Probably there was a bump in adoption once the first Segwit wallet software (Armory) was released, but that's gonna be a relatively small wave of enthusiastic and confident power users (I think Bitcoin Core can also use Segwit with CLI, but that's gonna make up a fairly small proportion of overall Segwit users). Electrum have now also released Segwit wallet software, but there's a limit to how much transacting can go on between early adopters. More time is needed until a majority of users and services are using Segwit as standard practice.
Have some big exchanges or payment providers stopped using Segwit addresses?
Most services haven't switched over to Segwit addresses (many are waiting for the Bitcoin Core implementation to be released). So I don't think many of them have been using Segwit since it activated.
