Does ShadowBot manage claiming my rewards?
Yes, ShadowBot automatically claims any outstanding rewards you have earned without you having to do anything. You can not turn this feature off, but why would you really want to?

https://shadowbot.us/shadowbot/faq.php#does-shadowbot-manage-claiming-my-rewards

I guess auto claiming rewards versus not auto clamining rewards is a thing between homo oeconomicus and convenience. Right now it is purely on the homo oeconomicus side.
Grabbing the rewards as soon as possible means a higher SP amount which results in bigger votes for the posts the account is voting for in the 2h rhytm. It's unpossible to maintain that efficiency with manual work. It clutters the wallet transactions list, but thats more a failure of the steemit frontend not to be able to filter that.

But, they announced to add a setting to be able to change that function with the next upgrade. So with the upgrade "Elysium Fields", you can change that. Dunno when it will be.