"οὐδὲν ἄλλο ἢ ..." is an expression meaning "nothing else than" (III.2), and it's followed by a genitive of comparison. There actually is a relative pronoun in your bolded section: οὗ, a neuter genitive. (Note the rough breathing and the accent; it's not οὐ or anything.)
You would expect this relative pronoun to be in the accusative because of its position in the subordinate clause, but as the "οὐδὲν ἄλλο ἢ ..." construction expects a genitive, the accusative semi-regularly becomes that genitive through case attraction (§2522), with elision of the antecedent. Without case attraction, and with the antecedent made explicit, the sentence is effectively:
εἰσὶν δὲ οὗτοι οἱ οὐδὲν ἄλλο οἰόμενοι εἶναι ἢ τούτου ὅ ἂν δύνωνται ἀπρὶξ τοῖν χεροῖν λαβέσθαι
They are those who believe nothing else to exist than that which they could grasp firmly with both hands
ἄν + subjunctive just expresses potentiality here.
For the second part, "ὡς ἐν οὐσίας μέρει" isn't rendered very literally in your translation; making it more explicit:
[...] οὐκ ἀποδεχόμενοι ὡς ἐν οὐσίας μέρει
those who don't accept [...] as (being) in the category of existence
ὡς is just the standard adverb of manner, or however you want to classify that exactly; different dictionaries and grammars call it different things, but the function is straightforward.
