You can use the Feature Vertices to Point tool in Data Management toolbox to create points at the beginning, then snap your ends to those points. Alternately it would probably work to create a duplicate of your line file and snap to that.
I'm probably missing something and if so I apologize but it looks like you have two different "snap" issues. In the case of manual editing with lines, if you are adding to a line with the sketch tool you need to have The Edit Sketch box in the snapping environment checked to get it to snap to itself, but if you are selecting and moving a pre-existing vertice then up in the Layer window you need to have your layer's Vertex or End box's checked. If the case of automatic cleanup, you might consider using the Integrate tool, which functions a bit like the coverage clean in terms of making vertice's coincident. Do it on a copy since the changes are made in the layer instead of producing a new one, and you may want to play around with different tolerances to get your vertices to "snap" without moving others too much. One other thought, there are snap, sticky move and tolerance settings that affect your work, and if one of them is set large you have to move your vertice beyond that tolerance to get it to move. The Editor - Options settings for snap and sticky move are in pixels and hence how closely you are zoomed in can affect your snap/sticky "distance".
All the tools you have described are designed to work against other features. Its easy for a human to make the call on what the two end points should do when you're trying to trim or snap dangles from the same feature, but less so for tools or a topology fix that has no guidelines. This could be improved.
You could try the Feature To Polygon GP tool which will create polygons from your lines and snap dangles with a tolerance. In the case of undershoots the two ends are moved to a midpoint. If you need to keep the building lines, you could then use the Polygon To Line GP tool, which means you wont see overshoots.
Thanks for the idea Trish. I've spent so much effort trying to make the proper tool function that I wasn't thinking of workarounds.
I tried both of your suggestions and still no luck getting the topological error tool to snap to the extra features. I can get it to snap to a feature 10 metres away on the same layer and the addtional layers from your suggestion, but not the vertex that's less than 1 metre away. It keeps returning "Snap dangle error failed." It's almost as if it searches, finds the other end point of the current feature first, which it can't snap to, then returns the error before searching and finding the vertex of the additional layer. If I create a new seperate feature within the specified snap distance I can successfully snap to that using the tool. Just a bug witht the Fix Topology Error Tool I suppose.
Thanks for your suggestions and time.
Scott