Ok, firstly Trisplitting:
Get a scrap of paper and draw a triangle, that's your poly - then put a dot in the middle of the triangle, and draw lines to each corner of the triangle, you now have 3 triangles where there used to be one.
I'm not sure how you handle your model data, I stored my vertices, then had a list of polygons with 3 variables for the vertice numbers, then the 'renderer' would genereate the 3D model from the polylist, which would refer to the vert list to get the coordinates. So I could easily share vertices between polygons. It would be possible to delete a polygon without leaving a hole too - you just make a new vertice in the middle of the poly, like the tri-split idea, but you get the vertices the polygon uses - then they're all replaced by the new middle vertice. By replacing I mean any polygon that uses the vertices, will use the new vertice instead - like pinching the polygon downto 1 vert - removing the poly while at the same time closing the hole.
By mirroring I mean taking all the verts on the left hand side of the scene and copying them along the x axis. Most character modellers work in halves, there's little point in duplicating work, so taking half a head and hammering the basic shape into it is a good time-saving technique, you just mirror the head so it's whole. The same is true of most bodyparts. Imagine a character mesh, cut in half down the middle, then each vertice that is <0.0 on the X axis is copied with an inverted x axis. You can be clever by automatically welding the middle verts - vertices close to the middle of the scene might be important, like a seam going down the middle of the face - if a vertice is close to the middle, don't mirror it. A completely automatic mirroring tool would be awesome.
By welding, I mean you replace the instances of one vert with another. Say you had 2 polygons making a square - you'd have 4 corners, 2 of them would have 3 lines comming from them, some would have just 2. The corners with 3 lines would take up 2 vertices, which is not ideal when modelling - so checking for vertices in very close proximity and optimising them so that duplicate vertices are removed is a good idea. Welding also connects the mesh for things like surface normals, a surface normal calulation will take the connected polygons into account, so welding includes all the verts in the calculation (which will mean smoother normals).
IMPORTANT: UV coordinates should always be checked when removing a vertice, if it's UV data is unique it cannot be deleted, the UV data should be compared between the 2 vertices when welding, if the UV data is identical then the weld is fine. There should of course be an option to overide this - but a warning of some kind when UV data is at risk would make things clear.
Van-B

Next time he runs past, GRAB HIM!