First, it's important to consider whether the company selling this software is the exclusive copyright holder or if they are themselves licensees of someone else's AGPL license grant. If they are the sole copyright holder, they may offer as little or as much of their code as they want under the AGPL purely out of the goodness of their hearts (or according to whatever business goal this actually fulfills). Under no circumstances would they by beholden to the requirements of their own AGPL grant. They may proactively give you updated versions of their own code for as long or as short a time as they please. It's their code.
On the other hand, if they are licensing the work under the AGPL because their code is a derivative of someone else's AGPL-licensed code, then they have an obligation to offer the code to anyone who "interact[s] with it remotely through a computer network". If you interact with their modified work through a network then they are required to notify you of your rights under the AGPL, and you are entitled to receive a copy of the code under the AGPL as part of those rights.
If they have such an obligation and fail to meet it, they are liable for copyright infringement since they are operating outside the terms of their copyright license grant, and the original copyright holder has standing to take legal action against them. (You may wish to notify them, and the Free Software Foundation, which has an interest in helping authors enforce the licenses they publish.)
All that said, here are the answers to your specific questions:
Is it "compatible" with the AGPL to only allow people who pay to have access to your source code?
Yes, you may do so as long as one of these is true:
- you are the sole copyright holder of any AGPL-licensed components, or
- you offer the source code to anyone who gets the binary from you, or interacts with your modified work over a network. (Note that you may take pains to ensure that the set of people who do this is strictly limited to paying customers.)
Can you actually say "you only get updates for one year, then you have to pay again" with an AGPL application?
If you're not the sole copyright holder of the AGPL components, you must offer the source code of the work when a users gets a binary from you or when a user interacts with the work over a network (if you modified it). If you are not the sole copyright holder, you may not say "I previously offered you source code for an older version; I am now offering you network access to a new modified version or a copy of a different binary and refuse to give source code for this current version." However, if you are the sole copyright holder, or do not give out access to later versions, then you have no obligation to offer the source for later versions.
If I pay for the application, am I allowed to freely redistribute it and/or use parts of its source in my own code under the same terms as I would with a free-of-charge download? (In this case, what's the point of locking the code behind a paywall?)
Any recipients of an AGPL-licensed work have the freedom to redistribute the work under the terms of the AGPL, whether or not they paid someone money to transfer them a copy. I speculate the point of charging for the code is that people (or likely, businesses) who pay for it are really interested in getting the latest updates immediately, rather than trawling around looking for someone who has a copy of the latest version (and having to trust that it hasn't been maliciously modified by this secondhand reseller).