Is there any wallet where I don’t have to worry about what network I’m connected to and what chains I have ETH on? 

I want to seamlessly experience an app without worrying about any of those details.

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The former, but taken further.

Say I have Base ETH in my wallet. And there’s an app I want to use on Blast or Degen Chain. I don’t want to have to think about bridging. Is there a wallet that would handle bridging and interaction in the background?

do you mean a wallet that lets you connect to apps and spend ETH on any chain you have it on? Or just an amalgamated view of all your ETH across chains?