Two months late, but still...
First of all, IANAL, ask a real professional, and all the usual stuff.
I'll start with 3.
You're touching something very big. According to the FSF, not just AGPL, but even GPL forces you to open-source anything that is dynamically linked to a (A)GPL binary.
According to some other people, you're not really forced to do it. There hasn't been any major lawsuit that can confirm either view of the problem, but the spirit of the GPL is that you should open-source the client application in its entirety. Needless to say, this goes to AGPL too, as AGPL is more restrictive than the GPL. Important note: you can license the rest of the client as GPL, not AGPL, since AGPL allows for this.
1 and 2 are virtually the same, to a certain extent. In both cases I believe you're allowed to do what you want under AGPL. Let's look at the relevant text in AGPL:
Notwithstanding any other provision of this License, if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software. This Corresponding Source shall include the Corresponding Source for any work covered by version 3 of the GNU General Public License that is incorporated pursuant to the following paragraph.
So, if you modify the client library that is AGPL licensed, you must provide the modified source code to all parties, that will communicate with it, that is to all servers. As far as I understand, you're not modifying the source code of the client, so it's not your problem at all.