Earlier quoted context omitted.
I think "gaining half a bit of information" about something can correspond to stuff that lets you update your probability distribution about it? I'm not sure.
yes, but your gain is still an integer number. the non-integer ratio is just a statistic (gain per number of units).
Entropy (in bits) is defined as - \sum_x (p(x) log_2 p(x))
There is no reason this has to be an integer, since probabilities are not restricted to being reciprocals of powers of 2.
Consider also that you can simply use a different logarithm base to get a different unit (e.g. use the natural logarithm to obtain the entropy in nats). It would be bizarre if the arbitrary choice of 2 as the base gave a unit that was indivisible.
I think this whole confusion comes down to the difference between a bit as a "unit of information in the sense of information theory" [divisible] and a bit as a "single physical one or zero" [not divisible]. The relationship between the two is that the entropy of a random variable is a lower bound on the average number of bits required to represent it.