Would it be difficult to find another remote job? Like I said, they're becoming more commonplace. Also, if you already worked remotely, a remote job is easier to get because they're already experienced in working remotely.
And no, you need a solid fast connection to work remotely. You need to be able to video conference, regularly download updates to packages and dev software that can be up to a gig for each download period. Then there's remotely debugging issues on a cloud hosted instance.
I've tried working remotely on a slow connection and it was a miserable experience that took ten times as long to do most things.