Think about it though, it sounds like you're using an Amiga with a spec from 1992 or thereabouts. No PC would open an archive just by clicking on it then either unless you installed extra stuff - in fact, some PCs wouldn't click at all back then ;-)
Although progress has been slow, the Amiga has come a long way since then too. I can click on an LHA and see the contents on my machines, but that's just because they're all a bit more up-to-date than a plain A1200. If you had OS3.9 you could simply double click them to open the file with the built-in unarchiver. With a little bit of perseverance though you could at least extract an archive tool by hand, and once that's done you could use that from then on.
Unfortunately, by default the Amiga doesn't recognise files that have no icons associated with them. There are patches you can install which give it this ability, then it will act more like Windows etc. Most functionality that you seem to be looking for wasn't standard back then, but can be added by installing third-party apps from Aminet and other places.
For starters, you could do far worse than install Directory Opus 4, it's old but very good software for managing files, has built-in unpacking functions for archives, and filetype recognition and viewers so you don't need to worry about creating icons for "foreign" files.
It can be downloaded for free here:
http://dopus.free.fr/download.phpthough you will have to unpack it by hand using the Shell command given a few times above...
