Reseeding a PRNG with the generated PRN Would reseeding a PRNG with the PRN from the previous generation have any effects whatsoever on the quality of the generated numbers; if so, does this depend on the PRNG used, or are the effects generally applicable to any algorithm? 

Perhaps I'll enhance the question with some pseudo-code:

firstSeed := get_seed_from_entropy_pool()
seed(firstSeed)
firstPrn := rand()
seed(firstPrn)
nextPrn := rand()


Is nextPrn already of inferior quality? Would quality decrease if repeated?
In most PRNGs, this leads to a total disaster, in particular when the generator's state is bigger than the result returned as pseudo-random (call to rand()).

If you take Mersenne Twister with its big period guaranteed and apply the construction you expose, as it returns 32-bit integers, eventually you will get a repeated value that was used to seed previously, so it turns into a loop. You'll get an about $2^{16}$ repeating sequence of pseudo randoms, according to the birthday probability, and not the guaranteed generator's period length. This same issue applies to all other generators if it's seeding is deterministic and guarantees that a given seed always generates the same sequence of pseudo randoms.

On the other hand, if you do some steps of autoseeding first and then you use the PRNG without seeding it anymore then it shouldn't affect its properties.