I agree with the general idea of what you've put forward, but not all the details.
I think the "trigger" should be more along the lines of
Reputation <= -N, rather than Downvotes >= N.
I visit "Selected Best Nodes" every so often and see nodes with numbers like
Rep: 185 (+200 -15): I wouldn't want those automatically hidden.
I've noticed others have discussed this point and -10 has been mentioned a few times:
that would seem to be a reasonable figure to me.
I'm not sure about displaying the vote count on the node.
If I haven't voted on the node yet, would I see both the current votes (as I currently do after voting)
and the radio buttons to allow voting (as I currently do before voting): it could be somewhat confusing;
especially to someone fairly new to the Monastery.
There's also the issue of implementing such a change and how much work that might entail (I've no idea on that).
As an alternative, can I suggest simply adding a link to "Worst Nodes" as part of your posted message
(i.e. "This node was taken out ..."):
that stands a good chance of having a current, or at least a fairly recent, vote count;
it would also highlight repeat offenders.