V1 is the maximum speed at which you plan on rejecting takeoff should something go wrong, other than in situations where you have serious doubts about the aircraft's ability to fly e.g. catastrophic fire or structural failure before rotation. It is also tends to vary quite a bit depending on operational conditions. Aircraft manuals will have many tables, charts, and calculation instructions like the one above in order to allow pilots to calculate what their V-speeds will actually be on any given day... though, admittedly, this is usually handled by flight planning software there days.
Generally speaking, V1 is dictated by how much runway you require to stop and how much you have available to you. A lower takeoff weight or a lower density altitude will both serve to bring Vr down, which, ceteris paribus, can bring V1 and Vr closer to each other. A longer runway gives you more room to stop, so it can bring V1 up.
On a balanced field, V1=Vr so you have to option to reject the takeoff right until you're actually rotating. To look at it another way, you have as much time as you could possibly ask for in order to decide not to take off if you encounter a problem during the take-off roll.
As for Vmcg, this is the minimum speed at which you have sufficient control authority on the ground to overcome the adverse yaw you experience with the critical engine inoperative. It would only really come into play to make sure that you don't back yourself into a corner where, if you were to somehow have an engine failure, you also wouldn't have enough airspeed to be able to overcome the asymmetric thrust from the still-operating engine(s), so you wouldn't be able to continue to accelerate and take off safely, but are also going too fast to be able to stop within the available runway distance. Honestly can't say I could think of a situation where something like this could be an issue with anything but the most off-the-wall back country and bush flying these days.
FYI, in case I wasn't clear enough, an engine failure at or above V1 is not, by itself, grounds for rejecting takeoff. That's where your V2 number comes in. You rotate at Vr and climb out at V2.
The only time you would reject a takeoff above V1 is if you think that the chances of dying are greater if you get airborne than if you run off the end of the runway.