As someone who has done some stained glass work, cutting glass is not difficult, as long as it is not tempered glass. It can take a little practice. This link shows how in pictures. Even better, this YouTube link does a very nice job of teaching you to cut glass.
Use a straight edge to guide the cutter. Better glass cutters have an oil reservoir, used to lubricate the cutting wheel. If not, you can also put a drop of a thin oil on the cutter before a cut.
Make the cut in ONE firm smooth stroke, not in multiple passes. (This is what takes the practice, to know how you need to press down on the cutter, while still making a straight line. Don't push too hard, just firmly. You will usually hear the sound of the cutter making tiny splinters in the glass, and you can hear if you got it right.)
Once you have made that one firm pass with the cutter, it is time to break it along the scribed line. With a straight line to cut, you can often take the glass in two hands. A bit of pressure, trying to bend the glass against that scribed line will usually cause it to break.
Sometimes the glass just won't break with a firm hand along that line. If not, then turn that glass cutter around. Not that the butt end of the glass cutter probably is shaped in the form of a small metal ball? This is to help you to break the glass along the line you just scribed.
The trick is to hold the glass with one hand partly off the table, with one edge still on your work table, and your fingers spanning the cut line. Now, with your other hand, tap the glass LIGHTLY with the ball end of the glass cutter from UNDERNEATH that cut line. You will hear/see/feel the glass suddenly break along the score line. If the crack does not travel all the way, a few more light taps with the cutter will do the job.
Since the table is supporting both pieces of the glass, when it does break, nothing bad will happen.
Be careful handling glass! If you are not comfortable with it, then a thin pair of gloves will not be a bad idea. I would also recommend not doing this work in your living room or kitchen, as glass splinters are nasty and hard to find. Sweep up carefully afterwards.
As far as cleaning goes, I'd be surprised if the glass is permanently stained, but that is possible. If you are just using it for a bit of a greenhouse, a bit of stain is not a problem anyway. I'd try some Windex first (or paint thinner, or a citrus based cleaner.) A razor blade is also a common solution, but you can scratch glass with a steel blade if you are not careful.