Stabilizing your craft in orbit

Someone else had a very similar problem to the one that you're having right now. Basically, your probe body is running out of juice some times. But there seems to be more to that in your case.

So the first part is rather easy to resolve. Once you're in orbit, right click on the solar panels and then click the extend button. When in sunlight, this will provide power. You can also set up action groups (which are triggered by the number keys) to toggle your solar panels. This is done from the VAB or SPH. It's a good idea to also have some battery power too, as you're not always going to be in the sun. Another option is to completely forgo solar power, and use RTGs instead.

As for you second problem, it seems that either you built your rocket asymmetrically, causing it to spin, break up, and then explode, or you're using too much thrust, causing it to break up, spin, and then explode. Obviously, ending in explosions is not a good thing in this context. The solution to both problems is basically the same: use the symmetry tool when building, and add enough bracing using struts. In fact, struts are so light it's usually worth it to over-do it on the struts.

Even a 0,1 m/s difference in speed makes a difference over the years. The only way to ensure that they stay in their position is to fiddle with their orbits in the save game file.

Note that this only works as long the satellites remain in their on rails-simulation, so you can't directly control them by using the control center. Otherwise they will always drift.

You might find these links helpful, there are lots of tutorials how to do this:

Scott Manley's video tutorial

Orbits in KSP wiki