Shine, if I read your description, you save 3 or 4 different versions of the same file psd, jpeg, tiff and web. My first question is - why? I do most of my editing in Lightroom, onOne Photo Suite 9 and Nik Software (now Google). I always save my original, raw file - one of the really nice things about Lightroom is that it doesn't make any changes to your "master" file. Rather, it saves all the editing in an .xmp, or "sidecar", file and only applies the edits to the image if you export it to something like PS or the web. I often will make changes to my Lightroom edit using Nik's software or onOne and these are saved as tiff files. I am also very fussy about keeping an image. I use the Library module of LR to go through my images after importing them and get rid of any that I know I will not be keeping - things with parts that are OOF, bad shots (e.g., an experiment that didn't work), etc. I keep very few images as jpgs or tifs. If I need a jpg for web posting, then I will make it when needed, not before. All my library images are kept in the raw form, until I need to convert it to something else. I currently have about 1.5TBs of images in my library. I could probably cut that down a bit, if I went through and culled some of the jpgs and tiffs that I don't need anymore, I can always remake them.
The other nice thing about LR, is the ability to make Virtual copies and work on them without making actual duplicates. Lightroom looks after all the "housekeeping" for the virtual copies. With these virtual copies, you can try out all kinds of different effects to see what they look like, even making a black and white conversion without requiring a lot of storage space for these "copies".
Another question, is why do you make a tiff for printing? You can print the edited raw file, which is what I do, unless it is already a tiff from one of the plugin packages. When you print, LR simply apples all the "edit changes" from the xmp file before it is sent to the printer, at least that is my understanding.
As far as how to organize these, I make extensive use of LR's ability to "keyword" all my images. I keep all images in a date based system. The top level is the year, then the month, then the day they were acquired. With keywords I can probably find just about any image in less than five minutes, assuming I know the key words.
I also make extensive use of collections, but that is another topic for discussion some other time. Anyway, I hope this helps. I will be travelling for a few days and may not get back to this discussion for a day or so, but if you still have any questions, then post them and I will try and answer them the best I can. Hopefully, others will add to this forum with their thoughts and experience.