Here よく describes how something was done. The action being done is sleeping. You slept well. More specifically, よく is an adverb modifying a verb, ねます.
The reason why あまり and 全然 are not suitable answers is because they require the negative form of the verb they're modifying, i.e., [あまり・全然]寝ませんでした。If 寝ます was in its negative form, they'd be suitable.
えんぴつが 六本 あります。
When counting long, cylindrical obejects, all of the numbers end in -hon except the number 3 (which ends in -bon) and the numbers 1, 6, 8, and 10 (which end in -pon). See Puni Puni Japan's explanation of counting with 本 for further details.