I just checked, following this discussions, Even Shoshan dictionary, and low and behold: indeed /akhvishim/, as the speaker of the exercise says it. Surely very few native Hebrew speakers know it, and even less say it. We all say /akavishim/.
Yes. ἀραχνοειδής is cobweb-like (said of veins, urine, the retina in a medical context). But where English prefers taking the scientific vocabulary from the classical languages, Hebrew can make them of its own stock, being a classical language itself.
What do you mean by pronunciation depending on context? We had a discussion here about the plural of עַכָּבִישׁ, were there are colloquially used variants. Like in English, if you discussed, whether the correct plural of roof is roofs or rooves.