Recently, i've tried tuning from standard E to D tuning (i failed because the tunings kept changing). For some weird reason, my bridge (Ibanez Edge III) sank downwards without even manually lowering it. The feel of the guitar is also slightly different and my high E string doesnt seem as loud.

Anyone can help me? Is this a bridge or string-related problem?
you need to level the bridge by adjusting the springs in the back trem cavity.

there's a floyd tutorial and QnA at the top of the page.

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You are loosening the strings when you tune down. The bridge is spring loaded so the tension on the springs are loosening also making your bridge sink into the body of the guitar. I have done this many times. I would either take a spring out of the tremolo which is under the plastic cover on the back of the guitar or you would probably be better off taking it to a guitar shop and having it set up with the tunings you like. I bought a tremolno from ebay for the guitar I use to change tunings alot. It is about $45.00 and there is no modifications that have to be done to the guitar to install it. It pretty much locks the bridge but there is an adjustment on it also that allows you to use the tremolo too if you would like.
Edge Pro's and Floyd's tend to bitch about tuning changes because they are set to support a certain amount of pull from the strings. To little a pull or too much a pull will directly effect whether or not your bridge is level. It will obviously lower the bridge when the strings are loosened and raise the bridge when you've tuned up too much.

In short the guitar has to be set around whatever tuning you're playing in and it has to stay in that tuning unless you want to adjust the guitar.

If you are unsure about how to change the tension on the springs in the back of your guitar like jj1565 said, I'd highly suggest going into a store and telling them your tremelo is slightly floated or off, and that you are unsure how to fix it. He'll probably (given he has the time) show and explain how to fix the problem. It isn't too difficult, just extremely tedious.
^jj1565 is a chick or a robot. anyway TS listen to jj1565 she has the best instructions on this kind of stuff, but be aware that it will take you some time to get it finally in tune and properly set up and you will probably be annoyed at it afterwards.