As @StephenLead said, ArcMap will Reproject your datasets on-the-fly. What is most critical is that each of your Datasets have the appropriate Coordinate System set beforehand. The other critical item is that you set the Coordinate System of your Data Frame. There are differing opinions as to what this should be set to, but a good rule is to set it to match that of the majority of the layers in your project. This will reduce the amount of on-the-fly reprojection that has to occur.
The Define Projection tool is appropriate to Assign the projection to a dataset that has an undefined Spatial Reference, but where you know what it should be.
In your case, I think the issue is in determining what the correct Coordinate System of your Contour layer should actually be.
When you use the Contour tool in Spatial Analyst, the output data should retain the coordinate system of the input raster. If the Raster dataset did not have a Spatial Reference defined, then neither will the Output dataset, however, the coordinates will still be those of whatever the input Raster was.
This is the first item that you can test:
- Open a
New session of ArcMap. Do not set the Coordinate System of the Data Frame.
- Add the Contour Dataset. Once it loads in, the coordinates shown in the info bar at the bottom should show the coordinates in the layer. This is a characteristic of ArcMap in that it zooms to and sets the spatial reference of the Data Frame, to that of the First dataset that is loaded in.
- Compare the coordinates that you see in the window, to the known coordinates that you know should be found in this data.
- Perform steps 1-3 with your Raster dataset.
Once you have both sets of coordinates from ArcMap, you should end up with 3 possible conclusions.
- If your Contour coordinates match those of your Raster, and those of your Known coordinate range, then you know that they have carried through correctly, with no unknown reprojecting occurring. In this case, you need to check the name of the Coordinate System to make sure you have the correct one to match your Coordinates. Once you have the correct one, then use the Define Projection tool, to assign to your Contour dataset.
- If your Contour coordinates match those of your Raster, but these both do not match your known Coordinate range, then either your Raster was reprojected at some point, or it was delivered to you in a different coordinate space than you thought.
You then have 2 options.
a. Obtain a new copy of the Raster projected in your known Coordinate System. Then, recreate the Contour dataset.
b. Determine the coordinate system of your Raster. You may be able to figure it out by examining the coordinate range. For example, Latitude/Longitude coordinates will be in the range of: Y/Lat = 0 to 90 and X/Lon = -180 - +180. If you see coordinates in this range, then it is a Geographic Coordinate System. Look for coordinates that match those of commonly used Projections in your area.
- If your Contour coordinates do not match those of your Raster, then a reprojection occurred at the time you created your Contours. This could be a result of having the Environment variable for Output Coordinate System set. Here is the help file that discusses the Contour (Spatial Analyst) Tool, and lists the relevant Environments.
One thing I noticed in your question that may be a source of confusion is your use of Geographic Coordinate System (GCS), and Projection. The terminology in ArcGIS uses Coordinate System interchangeably with Spatial Reference. Within these, you might have a Geographic Coordinate System, which is based off a Spheroid with units of Latitude/Longitude, or you might have a Projected Coordinate System, that is based off of a Projection, like the Lambert Conformal Conic, that you mentioned above.
Here is the window in ArcMap for setting the Coordinate system of the Data Frame. Notice that you have the option to choose a Geographic or Projected Coordinate system. I have chosen the Geographic Coordinate System: GCS_GDA_1994
When you mention Lambert Conformal Conic, you are moving into the realm of the Projected Coordinate System, of which there is quite a list, as in this screenshot below:
I have selected one, which seems to have the parameters you are looking for, namely a Projected Coordinate system based off of the GDA_1994 Geographic Coordinate System, but using the Lambert Conformal Conic Projection. This has units in Meters.
Hopefully this discussion has given you some clue of where to find the problem. Your best bet in any case is to go to the source of your data and obtain the correct projection information from them, as well as datasets that are already in that projection, for you to work from.
For the future, here is another question asked on GIS.se regarding how to determine the spatial reference of a layer. It also contains links to ESRI documentation discussing the same. ArcGIS 10 projection question
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
The Define Projection tool updates the coordinate reference system metadata only. It doesn't change the coordinate values. The same thing occurs (metadata only) if you change the coordinate system via the data's property page in ArcCatalog.
In 2), a geographic (datum) transformation is required because the source and target geographic coordinate reference systems are different: Bessel 1841 and WGS 1984. First Bessel 1841 isn't a true geodetic datum (and thus geographic coordinate reference system). Bessel 1841 is an ellipsoid. Someone didn't bother to define the GCS/datum at some point in this data's history. So, ArcGIS has no transformations for Bessel 1841. Looking at the data--it's worldwide, and there are no GCS that work for the world and are on Bessel 1841. I think my updated recommendation is just redefine the data as WGS 1984.
If you can make an educated guess as to its real GCS, you should use the Define Projection tool to update the information and then try projecting it again.
You could also modify the World_Cylindrical_Equal_Area definition to use Bessel 1841 as its GCS so no transformation is required.
Another possibility is to add the data to ArcMap, set the data frame to World_Cylindrical_Equal_Area (change its GCS or not), and export the data by right-clicking the layer and choosing data, export data. In that tool, set it to use the data frame's coordinate system.