Is it "slough" or "slew"?

The phrase is definitely "a slew of". This meaning of the word slew comes from Irish slue "crowd" and has nothing to do with the verb to slay; the fact that the two words are spelled identically is just another accident of language evolution.

Fact
Indeed, slew is the past tense of slay. However, slew is also a verb in its own right that means turn!

In American informal usage*, the noun slew, as you rightly indicated, indicates a large number. As in your second example, slew of questions is not an uncommon phrase. Thus, the usage of slough in a similar vein would be incorrect. That slough and slew could both be pronounced in the same way is a possible reason for the confusion of the two in writing.

The noun slough primarily means swamp or bog, and it often has a dreary or negative connotation, e.g. Slough of Despond (John Bunyan's The Pilgrim's Progess), slough of moral filth, etc. It can also be spelled slew!

Opinion
Frankly, I do not at all think slough would be a better metaphor, as you suggest. Furthermore, I do not think the etymology of the [US informal] noun slew has any relation to that of the verb slay or its past tense.*

Addendum*
Interestingly, Merriam-Webster suggests the origin of the noun slew thus:

perhaps from Irish slua army, host, throng, from Old Irish slúag;

*According to the Cambridge Dictionaries Online