I have two raster tiff files one is a raster that I created from 2 Foot contour lines, the other raster file is a small area that was surveyed to half a foot accuracy. My question: Is there away to essentially clip the area out of the 2ft raster where my survey raster resides and then mosaic or merge the survey raster with the 2ft raster?
Yes you can. This method (used in arcmap 10.1) will answer your question, but there are other ways to do it
Alternatively, you could just jump to step 6 and manage the merge using Mosaic to New Raster's parameters for its Mosaic Operator. However, if you use the Blend option, I would suggest using steps 1-6 above, but have an overlap of the two data sets to help the blend transition.
Its always good practice to check the areas of the joins in the new mosaic to ensure that there no anomalies in height.
Mosaic To New Raster:
Clip (Data Management):