Personally, I think a mechanical keyboard is worth it if you use your computer every day and use it to type rather often. I won't go into discussion of the keys because everyone has their preferences. I've tried MX blues, reds, blacks, and am currently using browns. The best is to go into a store and try each one for yourself to see what you like and prefer.
As for Osu!, I don't feel as if it gives any particular advantage if you have a good rubber dome keyboard, chiclet, or mechanical keyboard, as long as you're used to it. It can make a difference for Osumania because most mechanical keyboards support n-key rollover. However, for the regular game mode, I do believe that a mechanical keyboard is more reliable in actuation force (the force required to push the key to activate it), and the speed the key returns to its default position to be pressed again. Some rubber dome keyboards are not as reliable in this aspect.
In terms of stamina, it depends again on the quality of the rubber dome keyboard. Mechanical keyboards have very standardized actuation forces, and therefore the stamina consumption is very reliable. For a rubber dome keyboard, users tend to press the key too hard to guarantee a key input. A good rubber dome keyboard is still quite steady in its actuation force, and shouldn't necessarily take more energy to press the keys.
For high bpm (220+), I would greatly suggest mechanical keyboards. The reliability you need to get good accuracy for a stream is much more important at high bpm, and while you can "float" on a mechanical keyboard, it's extremely difficult to do on a rubber dome. Floating is depressing the key just past the actuation point, and letting it rise a little above, then repeating. This is the fastest way to mash, and learning how to float well on your particular keyboard will allow you to single tap and stream at much higher bpm.
TL;DR If you play 220+ bpm streams, get a mechanical keyboard. For everything else, you probably don't need one, but I wholeheartedly recommend a mechanical keyboard if you use your computer daily.