It is worth noting that non alt process print will be as sharp as an inkjet print or a silver gelatin print. The reason for this is that the image is formed within the fibers of the paper rather than in a thin coating that lies atop the paper. That said, one can certainly make decently sharp alt process prints using a nice smooth paper. Any texture to the surface of the paper will diminish the perceived sharpness of a print.
As far as negatives go, realize that the contrast of your negative needs be tailored to the printing process in order to get the best results. With film, this means exposing and developing your negative with a particular alt process in mind. If you are open to a hybrid process, one can scan a film negative, apply an appropriate adjustment curve and print a digital negative to get around this limitation.
As for the various processes... from what I have seen carbon prints are probably the sharpest. I have no experience in making carbon prints but from what I have read/heard it is, in terms of technique, very different from most of the other alt processes and it has a fairly steep learning curve.
Cyanotype is very inexpensive and a good place to start an exploration of alt processes. In my opinion, the blue of cyanotype works well with photos of buildings (see below). As for toned cyanotypes, just be aware that toning comes at the cost of stability. I have a triptych of the same image in native cyanotype and two different toned versions all in the same frame. It has been hanging in our home for almost 20 years. The two toned versions are quite faded compared to the untoned cyanotype.
Gum prints are very nice. However, I would not put sharpness near the top of a list of their strengths. The environmental costs of using dichromate are high and properly disposing of the waste generated add complication to the process.
As for the silver-based processes... Yes, sliver is more expensive than iron, but it is still not that expensive in the grand scheme of things. Of course, I know nothing about your finances, so this is easy for me to say. If you want to go this route, learn the mechanics of alt process (coating paper, etc.) using cyanotype then switch to a silver-based process as the mechanic of many of the alt processes are similar.
Lastly, here are a few cyanotypes of building that I have made in the past couple of years...
Mill Building (Harrisville, NH)
Gosport Chapel (on a textured paper, so not really sharp)
East Quoddy Light (Campobello Island, NB)
Window Detail (Cyanotype Toned with Tannic Acid and Ammonia)