Don't worry about not having 2.05. As I recall, the only "feature" is the removal of narrator.device and its associated files. There are some other differences in the 2.05 Kickstart ROM that are only relevant to the A600.
For floppies, make sure to cover the extra hole on the disk with some sort of solid tape (or a taped piece of paper). This will make the drive think its a double-density disk instead of a high density one. But to work with it on your Amiga, you need OS v2.1 or higher or a product called CrossDOS (2.1 includes CrossDOS, you see). Mount the PC0: device (I think it's on the Extras disk in the Storage/DOSDrivers directory) and then you'll be able to read 720KB-formatted PC disks.
So your best option is probably
Amiga Explorer (part of the Amiga Forever package, and I think it's also available separately). It'll allow you to easily transfer files and disk images over a serial cable.
Getting your 500+ on to the modern internet is going to be a major challenge. Only one model of ethernet board was made for the 500, and I'm not sure I've ever seen one for sale. The only other ethernet option is a clockports adapter, a Subway USB card (out of production), and a USB to ethernet adapter. If you're willing to put up with dialup, a good old fashioned modem will get you online, but without a hard drive, more RAM, accelerator, and OS3.1, not much software is compatible with your machine.
Heavily-upgraded Amigas these days get expensive fast
