Does this vote get attributed to me? If not, who? I don't see it in my activity.
Automatic down-votes, like the ones for posts flagged as spam, and for questions closed as off-topic are given from the Community user.
Should I also 'manually' downvote a spammy answer or is that just a waste of my daily vote allowance?
Just flag it as spam; there is no need to down-vote it again, once you flagged it.
If, having found a spammy answer, I then notice via their profile that the user has a plethora of spammy answers, would working through them quickly trigger the "serial downvote/vandalism" script?
If you are flagging them as spam, the script that detect serial down-voting doesn't kick in.
Should I be working my way through a spammer's profile in this way?
If you mean looking for other spam left from the same user, then you can surely check if there is still spam that has not been detected.
Consider there are two types of spam: posts about something completely unrelated with the question (e.g. posts with links about boots), and posts about something related with the question that are promoting something (or that could be considered promoting).
In the first case, it is rather probable the account used to spam is destroyed, and all the posts made with that account are deleted. Still, flagging them as spam has the effect of moving them on the bottom of the answers given to a question; if then the spam was posted as a question, with enough down-votes the question is not shown in the front page.
In the second case, knowing how much of those promoting posts are written by a user is important for the moderators. If the user has contributed to the site, the account will not be destroyed (which is not even possible when the reputation is higher than a specific limit); it is probable the user will be said not to write such posts anymore, if they are considered over promotional.