The use of geographic vs projected is more up to you and your analysis needs. Based on what I've read so far in this question, it is not necessary for you to project into a UTM zone. However if you aren't consistent in picking the correct type of CRS, you will get alignment issues. Generally speaking (and there are exceptions) if your coordinate units are in degrees, you need a geographic coordinate system. If they are in feet, meters, or anything other than degrees, you need a projected coordinate system.
Part of what may confuse you is that NAD83 is actually a datum, and you can see it as part of the names for both geographic and projected coordinate systems. In ArcGIS's dialog they are grouped into folders. When I search for "North American 83" I get no results, so I'm not sure what options you would have been presented with to choose from. When you first see the dialog there should be two folders - one for Geographic Coordinate Systems and one for Projected Coordinate Systems. If you expand the GCS folder, and then the North America folder under that, you'll see the NAD83 that you want. Note there are several variants as well, such as CORS, which are updates/revisions to the original NAD83 definition. But you can ignore these unless you know you need to use one.
I would go back to your original tabular data and redo the conversion from CSV to points with the Add XY Event tool. In that tool, make sure you select the correct GCS system for the coordinates. Then export those layers to feature classes/shapefiles. At that point each of your four files (one and the three parts of two) should have the correctly defined coordinate system.
If you then add them one at a time to ArcGIS, whichever you add first will become the dataframe projection. All the ones after that will automatically be reprojected on-the-fly to that first CRS. Note that because you're changing datums you'll need to select a transformation as mkennedy mentions. This is separate from picking a CRS, though done via the same initial dialog box (look for the transformations button), and if you don't select a transformation at all or choose the wrong one, that can also result in a misalignment. If things don't line up at this stage, there is an error in which CRS you selected - either because something you selected was wrong or because what they're labeled as in the CSV is wrong.
If everything looks like it lines up correctly, you can then use the Project tool on the non-NAD83 layers to convert them to that CRS. Alternatively, if your dataframe is in NAD83 you can right-click on them and choose export, and in the dialog switch from the source CRS to the current dataframe CRS option. When that's done you should have all four feature classes/shapefiles in NAD 83 and you could merge them or add your xy coordinates for whatever.
Best Answer
As commented under the OP, my suggestion is to use the Point Distance geoprocessing tool, if you have Advanced licensing.