If I were to restrict background data on my phone to prevent data overage, will this slow my apps down?
No. It will just use wifi to do all the background stuff, rather than cellular. If you are USING a particular app on cellular, it will function normally. If the app is not being actively used, it won't be connecting and updating in the background unless you are on wifi (like a weather app, for example).
This article may be helpful: How to Monitor (and Reduce) Your Data Usage on Android
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