Why do my Shapefiles not align correctly?

If you want to assign a CRS to a displaced shapefile layer different to the current (false) one, do not Save As ... . This will reproject the data, but keep the displacement.

You should better use Vector -> Data Management Tools -> Define Current Projection. This will let coordinate values untouched, but change .prj and .qpj on disk. To see the change of displacement, you have to move the map a bit to force a redraw. This may be considered as a minor, confusing bug: http://hub.qgis.org/issues/7257

Rightclick -> Set CRS for layer is only working for the current QGIS project by writing the new CRS into the .qgs file, so will also not be what you want.

For the second question: The second file I found at opendataphilly.org, it is correct with EPSG:2272, and alignes well to openlayers plugin Openstreetmap background. If the other (from where?) should be in WGS84, you could add it with that projection to the canvas. By saving or defining it to EPSG:2272 you must have done something wrong.

This can be a really frustrating problem when dealing with spatial data. I'm sure many people on this site can empathize with your troubles!! My first thought is that it looks like you have the "PhilaCopIncsJune302012" file set to Penn. State Plane (ft) when it should be in WGS 84. WGS 84 is technically not "projected," as it is a Lat/Long system. What I would try is starting over with a fresh map and un-tampered-with data and setting the projections again to see if that fixes it. If that works, save a Penn State Plane version and see if that works for you.

Another possibility is that you have accidentally turned off on-the-fly projection and not noticed. I've definitely done that before.

If that fails, try looking at where the features are in relation to each other. This can tell you a bit about if one is in the wrong projection. For example, in UTM, if one feature is due south of the other by a lot, you probably accidentally selected the Southern equivalent of the UTM zone. If it's due east or west, you may have the wrong zone number. Once you do get everything working correctly, I definitely recommend using the same projection and datum for all of your data, especially if you are going to derive measurements from them. This will prevent a lot of headache (and heartache) down the road.