Can't see your photo so not sure about the corrosion. Typically, corrosion around battery poles can be removed with a solution of baking soda and water.
Secondly, wtf is a friction sheild (tm)? Odds are you don't need that.
After an oil change, if they spilled some oil when topping it up it could have leaked onto the exhaust manifold and burnt, causing some smoke.
Not sure what to recommend other than learning how to change your own oil, hopefully Mr. Lube would stand behind their work if they damaged anything but who knows these days.
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