Is the biggest star in a constellation usually labeled with the Greek beta symbol?

Answer 1

No, usually the brightest star is labelled alpha but there are a few exceptions e.g. Orion. The brightest star is Rigel but that is labelled Beta, while the next brightest, Betelgeuse, is labelled alpha.

Answer 2

No, this is completely false.

The BRIGHTEST star in the constellation will, slightly more often than not, be labelled "alpha". However, the guy who initially gave out the designations was doing it "by eye", so he had some fairly broad brightness bands, and if several stars were ABOUT equally bright, he tended to label them in order from west to east (though even that isn't an absolute; sometimes he labelled them according to some distinction which is not readily apparent). This happened (for example) in Ursa Major; all the stars in the "Big Dipper" part of the constellation are pretty bright (all but one are "second magnitude"... absolute magnitude between 1.5 and 2.5), so from the tip of the "bowl" all the way down the "handle" the stars are alpha, beta, gamma, delta, epsilon, zeta, eta ... the one of these that's actually brightest is Ursa Major Epsilon, and the one third magnitude star of the group (Delta, the star where the "handle" meets the "bowl", and which is very noticeably dimmer than the others) was apparently given its designation because Bayer thought it was more important to keep the chain going than to label them strictly by brightness.

There's no relationship at all between size and Bayer designation, because Bayer didn't have any way of measuring the sizes.