As I already knew, "R" in German is pronounced as "R" in French, and "er" in "Bruder" is pronounced approximately as "Brude" without "R"... but the "r" in "mehr" is not pronounced as "R" in French, why?
If an -r is the final letter of a word, the it is usually pronounced similar to an short "a". Mehr is no exception to that.
The proper IPA symbol is different.
Again I'm on my quest to add some IPA to this site:
How Bruder is pronounced was explained here, so I'm not going to repeat that.
Now for mehr. In IPA you would write:
The difference is that in this case, we have a stressed e and not a schwa [ə] as in Bruder. [eː] is a vowel that is produced in the front part of the mouth, whereas [ə] is in the center. As you already know that the r is usually pronounced [ʁ], which is a uvular (back of the mouth) fricative, it is easy to see that the way from [eː] to [ʁ] is farther than from [ə] to [ʁ]. If something is very far away, you ususally end up somewhere in the middle. This is called assimilation. And [ɐ] just happens to be halfway from [eː] to [ʁ].
Another aspect that contributes is, that unlike French, in German there is not much emphasis on the last consonant. To compare with la mère:
So the French tend to more clearly articulate the end consonants, at least if there is an e written at the end. The e is not pronounced, but it still makes the consonant better audible. Also, apparently [ʀ] is the predominant pronunciation of r in French, which is a trill and not merely a fricative, so it is even better audible, but otherwise very similar. If the r is between two vowels in German, the same happens, so for die Meere we get:
I don't really know about people who roll their r (as for example in Bavaria), but according to Wikipedia and my own experience the [ʁ] variant is the one most widely used in Germany.
"Bruder" is really not spoken without the "r", except if you are talking some kind of slang. It may sound like "Bruda" when spoken in some regions, but this is just a local dialect.