We don't really have enough information to determine what the advantages and disadvantages of multiple devices or browsers will be.
For instance, will the ticketing or reservation system prevent one account from have two active connections to the systems? For instance, does the system recognize that your account already has an active connection to the reservation system? If so, does it return an error or just create a redundancy? Someone with a ticket currently could test this by attempting to start two reservation connections at the same time and report back how the system handles it.
As far as devices, we can say that the main potential disadvantage arises from competing for bandwidth. Since the connection backlog will likely come from Disney's side, this downside should be fairly minimal. Given the widespread reporting that the App has functioned better than the web, it seems like a no brainer to at least use a phone and a computer. You might see a benefit by disconnecting your phone from your home internet, but that's speculative.
As far as multiple browsers, currently you can log into your Disney account from multiple browsers simultaneously without causing an error. If this is also true for the ticket and reservation systems, then the main potential downside to multiple browsers is the stress it puts on your computer processor and your bandwidth. The main advantage of multiple browsers comes in when refreshing offers some benefit. Since refreshing has reportedly solved the Stitch ate the page errors, this is a potential advantage to consider.
Multiple refreshes may also help to grab reservations if the system gets close to capacity and cancellations start to become available. All the wording from Disney seems to indicate that they won't hold reservations for you as you attempt to process them. If they were holding reservations (similar to how ticketmaster holds tickets) while you filled in your information, then refreshing to compete for reservations as they get released becomes a huge deal.
All this said, my plan will likely include two browsers (firefox and chrome) on my main computer, access from the app on my iphone, and another browser window with a separate IP on a separate computer. The bottom line, is we don't have enough tech info to confidently say what should be optimal.