When you use Alt R to make a loop cut you get that purple loop of edges - then you click once and there will be an actual cut where the purple guides were. If you move the mouse you'll slide that loop and it will stay where it is once you click again.
If you want several evenly spaced loops you need to hold the mouse still between the two clicks, then change the number where it says Number of Cuts in the tool panel near the bottom.
Remove edge loops by being in Edit mode, Edges, and right click, while holding down Alt, on an edge that runs in the direction of your edge loop - that will select the loop. Then press Delete and choose Dissolve Edges.
Vertices are points in space with XYZ coordinates - for example, in the middle of the world the vertex would be 0,0,0. There is a vertex at both ends of an Edge- they define where the edge starts and ends. There are many things you can accidentally do that will duplicate Edges and Vertices - there are lots of shortcuts that you can accidentally press which do all sorts of things.
To find these double edges or vertices, press A until nothing is selected, then you can use Border select - press B and drag a square around the suspected edge or vertex. Right up the top of the screen it shows details about the geometry including how many edges or vertices are selected - if you only see one, but it says at the top that 2 are selected, you know you have a double one (or triple one). Reading the manual from the beginning helps get an understanding of all this.
About the last question, if you make one Alt R cut it will make it exactly in the middle between the other loops UNLESS you move the mouse between two clicks, in which case it will slide. If you finish that action and make a new loop it will again be half way between two loops. If you want to evenly divide a face with many cuts, delete the ones that are there and start again - click twice so you don't slide, then immediately enter a number in the Number of Cuts field.