What is a rangefinder style mirrorless as against SLR style mirrorless?
Basically, an external (cosmetic) styling choice. I'd place more emphasis on how the camera performs.
Regarding lenses, if say I pick this Olympus OM-D it means that I also have to find an Olympus lens?
Nope. As I mentioned above, you can also use Panasonic micro 4/3 lenses. And there are adapters for other types of lenses, but you will generally lose at least some, if not all, automatic/electronic control functions. (i.e. autofocus, aperture control, lens based image stabilization, etc.)
Also note that there is a difference between regular/standard 4/3 and micro 4/3. (Both Olympus and Panasonic make both types.) Regular/standard 4/3 lenses can be used with an adapter, but again, you may lose some functionality. There's a webtool at:http://www.getolympus.com/us/en/lenses/ ... source=web
If I get only one lens that is capable of zooming like Mark have mentioned Olympus 75-300mm, does it mean that it’s always zoomed because I don’t have a lens that is smaller?
Sort of. More technically correct would be to say you can't get any wider than 75mm. So, for example, to get a group shot of several people, you would have to back up pretty far, and that may not be possible indoors.
I see some options like 17mm 1:1.8? What does that ratio mean? For instance I see 2 options for the lens 75-300mm. One is 1:4.8-6.7 and the other is 1:4.0-5.6.
The first number is the focal length(s) and the second is the maximum possible aperture(s).
So the 17mm has no zoom; it's permanently fixed at 17mm. (This type of lens is called a "prime".)
The 75-300mm can zoom from 75mm to 300mm and the maximum possible aperture in the first case also goes from f4.8 to f6.7 as you zoom in. (In the second case, the maximum aperture is slightly "faster/brighter" (lets in more light) since it goes from f4.0 to f5.6 as you zoom in.)
Or Justin mentioned a 14-150mm lens which is similar to most telephoto lens of compact cams. Is that enough to have for my need (usual indoor scenes and zoom for stage plays)
Depends on how much light there is and how far you are from the stage. But my guess would be "no" because, as I noted previously, that lens is on the "slow" side. i.e. it won't do well in low light. (The zoom/focal length range might be OK, tho.) For low light, ideally you want a lens that's f2.0 or faster (smaller f number--f1.8, f1.4, etc.).
My friend also mentioned about slim lens called pancake. But I don’t think they are capable of zooming right?
Sort of. But there are exceptions. e.g. the Panasonic X Vario pancake lens
. Better to rely on the specs--as noted above, a zoom lens will list a focal length range (that Panasonic lens lists 14-42mm), while a prime lens will list just one focal length.
I also noticed that the OM-D has no flash, so it means I have to buy a flash to complete this?
If you need a flash, yes. But, for example, if you're far enough from the stage to need a super-telephoto lens, any camera mounted flash won't impact the picture, at all.
After I click on the shutter button, it takes awhile for me to be able to click it again ... I would like to know what is this speed called so that when I check on the specs of the camera I would know where to look? Is this called min and max shutter speed?
No, it's not called max shutter speed. That refers to the maximum amount of time the shutter can stay open.
It could be called shot-to-shot or cycle time, but unfortunately, I don't think this spec is usually listed (because it's dependent on several things--image format/size/resolution, buffer size, memory card write speed, etc.). Although you might be able to get an approximate idea by comparing the "burst rate" of different cameras. (Just make sure you're comparing apples to apples, or factor into your decision when you aren't comparing apples to apples.)
I actually don’t know how much zoom is needed
There are several field of view calculators on the net. You just need to guesstimate your distance from the stage. But as a starting point, if you are 50 feet from the stage and you have the aforementioned 14-150mm lens zoomed all the way in to 150mm on an Oly OMD E-M5 (2X crop factor), your picture will cover an area around 6 feet wide by 4 feet high. (I'm guessing your kid is around 4 feet tall?) Obviously, if you are closer than that, the area captured will be smaller, and vice versa, if you are farther back.
Or you could check the EXIF data in the pics you've already taken to see what the focal length was.