Three scientists have just published a paper that give some estimated answers to those questions. The lower bound for the estimate of total DNA in the biosphere is approximately 5 × 1010 tonnes (five followed by ten zeroes, if you're unfamiliar with scientific notation). This contains 5.3 × 1031 megabases, equivalent to the storage capacity of 1021 of the most powerful supercomputers. The Library of Congress has been estimated to contain a mere 3 × 109 megabytes.
A great deal of that information, it turns out, is contained in plants - around 3.65 × 1031 Mb. Although prokaryotes (various kinds of bacteria) are more numerous than all the high organisms, each contains somewhat less DNA than an average plant or animal cell; nevertheless, prokaryotes contain a total of around 1.6 × 1031 Mb. Animals, including, of course, us, contain about a hundred times less information than plants, at about 4.24 × 1029 Mb. That's rather similar to the amount of information contained in viruses, around 3.95 × 1029 Mb, and somewhat less than the amount of degraded or junk information contained in leaf litter, at around 7 × 1030 Mb. Still, as the paper's authors remark, it's interesting that the various classes of organism each contain, within a couple of orders of magnitude, similar amounts of information.
As for processing capacity, all of the DNA in Earth's biosphere is estimated to transcribe stored information into nucleotides at around 1039 NOPS (Nucleotide Operations per Second). That amount of processing power would need 1022 supercomputers, given that one of the biggest can process 1017 FLOPS (Floating Point Operations per Second).
All of that information, the paper suggests, gives one definition of the present carrying capacity of the Earth. It also defines the amount of information in a biosphere that gave rise to a species that was able to ask and answer such questions. Which poses another interesting question. Will the biospheres on life-bearing exoplanets need to be of a similar size, if they are to give rise to extra-terrestrial intelligence?