One missing bit is whether this is technically one product (like one codebase, even if large) or not.
If those are completely separate products then using Scrum I'd go in very short sprints (1-2 wks) and sequence development work. So two weeks project A, then project B, then C, then again A - maybe for two sprints, then C etc. In such a situation a single backlog makes no sense, separate backlogs should be kept for A, B and C. I know at least one team that works like this.
Whether you need more POs is rather a function of knowledge about products. Maybe you need someone for each project, maybe you have someone who knows well enough A, B and C to be the PO.
If different products then when you try to do it by taking different stories from different backlogs every sprint what you will end up with is split team. Naturally people will specialize in given project, also it will be very hard to have a good definition of done (are we done if we can ship new increments for A and B but not C this sprint?). If you can't sequence projects with short sprints, then I would look towards Kanban for trying to put some organization into this.
If this is one product/one codebase - then things are much easier. Even if the team will have to touch different areas of the codebase because of different projects they will be still working on the same products so all mechanics of Scrum will apply nicely. One backlog, one PO.
One downside of this to be noted is that people on the team will context-switch and there is a penalty from doing this no matter what process you use. Whatever process you pick should try to minimize this as long as possible (as long as business will be able to hold). Nice thing about Scrum is that it has this built in agreement with the PO that context switches can occur only at sprints border - in other words team will get 1-2 wks to concentrate before having to switch to another project.
Also, don't forget about all technical practices of agile. Unit tests. Automatic builds & tests. Code reviews. Clever use of repos. High standards re. quality. All those are a must in such a challenging environment.