As others have said, welcome back to vintage Mac-land.
In my experience, checkerboard patterns are commonly symptomatic of failed capacitors in a Classic II, just like the SE/30 etc. I don't think that RAM and PRAM batteries are going to help you much there (though I would certainly pop in at least 2x4MB 30-pin SIMMs, and I would renew the battery also just to keep the thing ship-shape). If there is no sound or only faint sound, also, that's confirmation of capacitor failure. Removing, washing and drying the logic board may provide a temporary fix, is a useful diagnostic measure, and is a necessary step in the repair process anyway. I'd try that in the first instance.
Apart from the backup battery, which tends to cost a fair bit retail, the capacitors can be repaired very cheaply and fairly easily, and the repair is certainly well worth doing to preserve a Classic II in good condition. Some of us prefer a Classic II for aesthetic reasons to any of the other Compact Macs, even if it is a compromised technological design. Cute and powerful "enough."
Having said that, System 6.0.8L flies on a Classic II. The model shipped with 7.0.1, however, and with 10MB RAM, it will also run up to System 7.1 surprisingly well. I personally really like System 7.1 on a Classic II, but would not go any higher than 7.1, and would forego installing Open Transport and other, later System 7.1 add-ons. The 7.1 Update 3, however, is a good idea for bug-fixes and such.
Networking is going to be a pain if all you have is the Classic II and a modern PC. The Classic II is really pre-internet in the sense that it was designed in a way which was oblivious to its development. You could go the modem route, but I would say that the best thing might be to find another, more capable Mac to use as a bridge to the world at large. I would recommend finding a PPC running System 8 or 9, to turn on filesharing, and solve your problems that way. Something like a Wallstreet PowerBook, especially if it has the floppy expansion bay, would not take up much room and would prove really useful if you want to get into retro-gaming and such.
To illustrate: I have a Classic II running System 7.1 that can (for instance) mount a 500GB share from an Appleshare IP server that runs on my home network on a PowerMac 8600/300, connecting over localtalk/PhoneNet. Yessiree. There are some small extras needed to do this by way of use of Localtalk Bridge or similar, but it works perfectly. Software can be downloaded to the share and installed on the Classic II that way. You can also connect to TCP/IP networks if you want, but I don't see the point of that except to say that you'd done it. A Classic II is really not designed for that task, and is best left doing what it WAS designed for, which is running productivity applications, B&W games, and the like. Those things it does with style.
In my experience, checkerboard patterns are commonly symptomatic of failed capacitors in a Classic II, just like the SE/30 etc. I don't think that RAM and PRAM batteries are going to help you much there (though I would certainly pop in at least 2x4MB 30-pin SIMMs, and I would renew the battery also just to keep the thing ship-shape). If there is no sound or only faint sound, also, that's confirmation of capacitor failure. Removing, washing and drying the logic board may provide a temporary fix, is a useful diagnostic measure, and is a necessary step in the repair process anyway. I'd try that in the first instance.
Apart from the backup battery, which tends to cost a fair bit retail, the capacitors can be repaired very cheaply and fairly easily, and the repair is certainly well worth doing to preserve a Classic II in good condition. Some of us prefer a Classic II for aesthetic reasons to any of the other Compact Macs, even if it is a compromised technological design. Cute and powerful "enough."
Having said that, System 6.0.8L flies on a Classic II. The model shipped with 7.0.1, however, and with 10MB RAM, it will also run up to System 7.1 surprisingly well. I personally really like System 7.1 on a Classic II, but would not go any higher than 7.1, and would forego installing Open Transport and other, later System 7.1 add-ons. The 7.1 Update 3, however, is a good idea for bug-fixes and such.
Networking is going to be a pain if all you have is the Classic II and a modern PC. The Classic II is really pre-internet in the sense that it was designed in a way which was oblivious to its development. You could go the modem route, but I would say that the best thing might be to find another, more capable Mac to use as a bridge to the world at large. I would recommend finding a PPC running System 8 or 9, to turn on filesharing, and solve your problems that way. Something like a Wallstreet PowerBook, especially if it has the floppy expansion bay, would not take up much room and would prove really useful if you want to get into retro-gaming and such.
To illustrate: I have a Classic II running System 7.1 that can (for instance) mount a 500GB share from an Appleshare IP server that runs on my home network on a PowerMac 8600/300, connecting over localtalk/PhoneNet. Yessiree. There are some small extras needed to do this by way of use of Localtalk Bridge or similar, but it works perfectly. Software can be downloaded to the share and installed on the Classic II that way. You can also connect to TCP/IP networks if you want, but I don't see the point of that except to say that you'd done it. A Classic II is really not designed for that task, and is best left doing what it WAS designed for, which is running productivity applications, B&W games, and the like. Those things it does with style.