Will using both Google Adwords Auto-Tagging AND Manual Tagging with UTM variables create any tracking problems in Google Analytics?

Manual tagging + auto-tagging (utm_term={keyword} along with autotagging) will create problems with the data in GAnalytics (duplication, etc) but if you use your AdWords parameters in the format of kw={keyword} then there won't be any problem. If you want help, set up a call.

As @Michael said, when you have your Adwords & Analytics linked, the initial data is set by the UTM parameters.

Then, a couple of minutes later, Google will import all of the data related to the GCLID stored in the session, and that will overwrite all of the data you originally had.

We've used this to our advantage with a client who the utm_campaign variable in their own custom tracking platform. We let them know they would be fine, as their platform would properly record the campaign variable, and then a couple of minutes later, the GCLID variable in analytics would overwrite it.

We haven't seen any data duplication, except for when (for some reason) the GCLID doesn't get stored properly, which has happened occasionally. Otherwise, it works out fairly well.

Hope this helps.

There's a high chance you'll get duplicate data and think you're doing much better than you actually are.
Using both manual and auto tagging will result in your links appearing in both Advertising/AdWords and Traffic Sources/Sources/Campaigns (or whatever they're called now).
I usually stick with auto tagging only.
Good luck!