Yes. 532nm is the typical wavelength of green that you see in a majority of projectors and they are DPSS. There are some new green diodes on the horizon and some sources at other wavelengths that are prohibitively expensive but, for cheap green - and even a lot of expensive greens, they're DPSS. If you're going to build a projector, you probably want to get something more than 5 mW as you can buy all kinds of off the shelf laser projectors at 5mW from companies such as Chauvet and American DJ. I don't what your need is. What are you trying to use it for? That will help determine what kind of power you need. If it's just for home use, you don't necessarily need much. If you have bigger plans, then you need more power which corelates to more money. When you get into modulation, there are two types - analog and TTL. TTL is simply on or off which means full brightness or nothing. Most of the greens you're going to come across at low powers - say under 100 or 150mW are going to tend to be TTL. If you want to really have control of brightness, then you want analog which usually puts you over 200mW. Yes, at that power, you "should" have goggles but, many people that are careful may not bother with them but, it's always a good idea to use them in theory. If you have aspirations of building a projector, you've got a ways to go and, this is NOT a cheap hobby by any means. The laser itself is but a small part of what you're going to be investing in. There is galvos, a case, power supplies, other electronics, optics, some sort of control software and equipment. The list goes on.... Eventually, you will come to appreciate that the Dimention from X-Laser I mentioned above
is a good starting point!
I see you've added your location and there are an awful lot of excellent members and resources right in your backyard. I'd seriously begin looking through Rob Stanley (stanwax's) web site.
http://shop.stanwaxlaser.co.uk/
I just noticed he's got some web site issues but, there is still a lot of good things to study.