Welcome @muffinDad 
Great questions here 
The Foundation Template is a little lacking in “savings tracking” functionality, and we hope to address that in the next year with some solid, but hopefully intuitive, workflows.
In the mean time, I’d recommend learning a bit more about the Savings Budget from the Tiller Community to see if it helps at all. It keeps track of savings per category month over month (e.g. tracking overspent/underspent in each category in each month and recording it as “savings”)
It’s not exactly the same concept as the sinking fund (at least the way you’re describing it, but I’m a bit unfamiliar with that concept generally) but it could help with not having to use a manual sheet.
As for the money moving between savings<>checking and not wanting it to show up on the budget as income, there are a couple ways to do it. You can use a category that’s set as a transfer type (those do not show on the budget by default) or you can use an income category and mark it as “hide” in the “Hide from Reports” column on the Categories sheet.
Don’t worry too much about getting the category setup perfect the first time. If you change your mind later you can use Find & Replace to update the category in the appropriate places (all sheets) or you can use the “Rename category” tool in the Tiller Community Solutions add-on.
Hopefully that helps!