Dumb question...

So lets preface by saying I'm new...

I'm curious, if i backsweten and then bottle carb my cider, will that raise the alcohol content from when I checked the SG-FG? I'm just wondering about the yeast eating up the new sugars...

Yes, the yeast will eat the sugars - its kind of their thing

Alcohol is not what you need to worry about - bottle bombs will be the problem.

If you back sweeten with a fermentable sugar, you will need to pasteurize the bottles when they reach proper carbonation. There is a sticky on the forum about pasteurizing under these conditions.

I attempted to stop fermentation in a small batch of wine coolers by just putting them in the refrigerator. All the cold did was slow the yeast down. The last couple bottles (it was a 1.5 gallon batch) were very foamy.

the amount of alcohol generated from the fermentation needed to carb to the standard 2-2.5 volumes will contribute around 0.25% abv on top of what you already had, more or less, going on the standard yet highly challenging '2 gravity points to carb' theorem.
watch them bottle bombs!