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It is shown in An average-case depth hierarchy theorem for Boolean circuits a random oracle makes $PH$ infinite.

Is it possible to also show $UP\not=NP\not=\Sigma_2^P\not=\Sigma_3^P\not=\Sigma_4^P\not=\dots\not=PH\not=PSPACE$ with respect to a random oracle or does a random oracle give $UP=NP$?

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Yes. Beigel CCC '89 showed $\mathsf{P} \neq \mathsf{UP} \neq \mathsf{NP}$ with probability 1. Combined with Rossman-Servedio-Tan, this gives the result you want. You should always try the Complexity Zoo for questions like this...

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