Answer:
The short answers are Yes, it's random, and Yes, it "waits" for some time.
Different tRNA's just float around in the cytoplasma, and diffuse more or less freely around. When one happens to bump into the ribosome, at the right spot, right orientation, and of course which has an anticodon matching the codon in frame of the mRNA being translated, it gets bound and takes part in the synthesis step that adds the amino acid to the protein that is being synthesized.
The concentration of the various species of tRNA is such that translation occurs in a steady fashion, but there is always some waiting involved for a suitable tRNA to be bound. In that waiting time, the ribosome and mRNA stay aligned - that's because the energy that is required to move the to the next position is delivered as part of the same chemical reaction that transfers the amino acid from the tRNA to the protein that is being synthesized.
I'm not entirely sure what happens if there is significant depletion of a particular species of tRNA, but I think it's likely the ribosome / RNA complex can disassemble spontaneously. But spontaneous disassembly can't be something that occurs very easily after translation was initiated, since we would end up with lots of partial proteins which I expect would be lethal very soon.
(Can't know for sure though, but it would be very hard to set up an experiment to measure just what will happen and even if you got a measurement it would be hard to figure out how it applies to normal, living cells. I can't imagine tRNA depletion occurs in normal, healthy living cells.)
