If outputting 2 bits leaks 2 bits of state, you're probably not using a cryptographically strong PRF.
In general, a N bit output will be inconsistent with 2^N - 1 of your 2^N possible internal states; the point of using ~256 bits of entropy is to ensure that it's computationally intractable to figure out which, and that only works if the attacker can't work out the first (second, etc) 32 bits separately from the subsequent ones.
0: of entropy, mind, not data. If the attacker knows your 64-byte RNG buffer, and you inject 2 bits of entropy, there are only 2^2=4 new 64-byte blobs that your buffer could have evolved to, so you have 512 bits of data with only 2 bits of entropy.
1: for any algorithm that is; on a per-event basis, there's obviously a chance that you happen to produce output that's consistent with multiple possible internal states.
Fixed, but I'm assuming you're actually using the RNG; if you only generate 2 bits of output per minute, you can get away with all kinds of dumb crap.
> So it depends on the definition of "block".
It's the same definition as any other (2)read syscall. Assuming a 32-byte read from /dev/random[0], it's equivalent to reading from a pipe with at least 32bytes buffered if the RNG is already initialized (read returns immediately); if the RNG is not initialized, its the same as reading from a empty pipe - the kernel switches to some other process after making a note to resume the calling process when some event happens ((2)write for a pipe, RNG initialization for /dev/random).
0: Or /dev/urandom, which should be literally the same character device, because there are never any legitimate reasons for them to differ.