At low powers, the Moon can be painfully bright, much like looking into bright headlights at night. It won't burn your retina,
, but it will burn a ghost image into your eyes that'll last a few minutes. "Moon filters" come in two colours, green (don't even go there!) and grey. The grey ones are usually refered to as "neutral density filters". I prefer to use two polarising filters as this allows me to set the light attenuation to what I want. I use these polarizing filters also on Jupiter and Saturn as they are very bright in my scopes.
Yes, you can see DSO's without filters, galaxies, open and globular cluster, and nebulae of all types. Nebula filters are designed to only transmit the specific wavelengths at which their component gases glow at, while killing the other unwanted wavelengths. They can be the only way you can see most nebulae well from urban areas. Consequently, they are no good for viewing anything else but nebulae, as everything else glows with the light of the entire spectrum.
Unless you have access to a huge scope, anything over 20", you are only going to see colour in stars. Everything else is too dim to sitimulate the colour perceptive cells in our eyes, so you'll see dim objects in shade of grey. Some objects, however, do glow in characteristic soft blues, and very faint green, but filters won't help bring out these colours any more than veiwing without filters. Just not enough light.
Today, yes, all filters should screw into all eyepieces. Only thing to be mindful of are the two typical eyepiece barrel sizes, 1.25" & 2".
The Lumicon page listed by Michael is brilliant! It explains well the various filter types, and what their best applications are, both for nebula filters and straight coloured filters, which are best suited to the planets.
Mental.