I'm curious about what happens when someone votes on a posting that is beyond the pay-out date. Does the creator still get reputation for the post or is the vote totally wasted?

If you upvote a post that is past the payout date, then the vote itself is accepted but nothing else happens. There are no rewards applied towards it but the person that voted on it does have their vote power percentage reduced (like any other vote would do.) I believe that no reputation is applied either in this scenario, but I need to double check on that to be absolutely sure.

This 'cutting off' payment seems rather counter-productive to me. Why shouldn't a content creator be paid by someone who finds, likes and up-votes an article even 10 years after it was posted? It shouldn't be that difficult to write code that pays out weekly but keeps paying out every week for the previous week.

There's been a lot of discussion since Steemit started over how long payouts should be left open (even indefinitely.) We used to have 2 payouts, one lasting 24-36ish hours (which activity extended the time on it) with the second being 30 days after the close of the first.

While I do absolutely understand the drive to be able to earn on posts forever (the long tail,) there is computing overhead that has to be dealt with too. Right now the vaaaaaaaaast majority of votes are placed within three days...and remember votes can only be given by steemit users, not just viewers. The witnesses (people who actually run the steem blockchain) have to keep all posts with active voting in memory however, making it more intensive with more open payouts. Right now we're sort of in that balancing act of what we'd like versus what's best for a quick, functioning blockchain. Especially when the processing time allowed for just about anything on here is at 3 seconds.

However, I do suspect over time we'll see the payout times be revisited to account for things like this.

Also; What happens when someone votes on an article that has the rewards with a line through the number (where the author is declining payment)? Does the curator earn any curation reward? Does the author earn reputation?

When voting on a post with a declined payout, the vote is accepted (with vote power % decreasing.) Technically curators still would earn rewards from their (up to) 25% or the payout...but since that's 0, they make a percentage of nothing.

I'll have to double check, but I do believe that reputation IS earned, even when payout is declined, while the payout period is still active.

What about articles that have been resteemed? When an article gets resteemed, I read that the payout deadline is extended so that the author has extra time to earn rewards from the resteem. What is the amount of time that is given for the extension? And if it is once again resteemed, is the time also extended again for every resteem period? If so, is there a limit to the extended time? For instance: Let's say someone resteems an article and it is extended by 2 days. Then someone sees the resteem before the extended time is up and resteems it on the 2nd day. Will the payout now be extended for 4 days? Can the extention be compounded every 2 days to extend the payout period indefinitely by people resteeming ever 2 days?

I believe that was probably from an older post before we started the 7 day payout. However, it wasn't resteeming that extended the time back then, but was voting activity with larger SP votes extending it more. This also provided some diminishing returns on the extensions beyond there being purely less SP accounts left to vote.

Now, it's a pretty hard closing time on payouts without there being any extensions. 7 days after posting, the payout is closed...with the last 12 hours (I believe) only allowing for downvotes. This 'downvote only' window was to provide time to deal with large SP users that were trying to 'game' the system.