Few questions about Quats

Finally added Quat support for my project, wasn’t that hard. Anyway, I understood how you can use 1 quat to define rotation, but what happens if I multiply this quat with another. As far as I understand, I must take in account that axis are affected by 1st quat, am I wrong? Then, when I create quat from given axis and angle, I calculate v, where every coordinate is given axis coordinate*sin(ang/2), I tried to normalize this v at the end of procedure & everything went wrong, without normalization everything is fine, why that so. Does that mean quat could be (x>1,y>1,z>1,w=any)?
And bonus question : are quats fater than glRotatef()? I know the difference should be around 0.1e10, but I’m qurios

I assume you are using quaternian multiplication. And that the order in which you multiply matters as in ab not eq ba you need to keep things normalised, are you sure you are normalising correctly?