Making a video showing the innards of the machine would be very helpful, or taking some more internal pictures. A lot can be diagnosed just by looking at the thing. But taking the Zorro-card out and having a look underneath is a good next step. Always roll back to stock, and that card is not a HDD controller. It does look like it has ZIP-ram on it (which might be causing trouble if bad) and it does look to have ports (possibly serial or parallell?).
You'll find that lots (most?) folks in the amiga-community do basic repair and maintenance themselves, and some basic steps are agreed opon by most. Rolling back to the most stock setup (no zorro cards, no HDD, no other addons (if there are any) and no external devices) is one of those. Next would be removing expanded memory. The A3000 has 2mb of ram soldered to the board (in all cases I've seen) and the rest of the ram either in zip or dip in sockets. Those can be pulled, and might very well cause quite a bit of headache if some are bad.
If stability-issues or other (graphical) glitches are present even with no Zorro and no HDD then removing memory would be the easiest next step. If problems are still present then more intricate steps are needed (like checking the graphicsmemory, custom chips etc.). But I really cannot stress enough the importance of having a really close inspection of that battery. If it indeed hasn't leaked yet and everything is perfectly fine then removing it is paramount. Corrosion might be very difficult to spot but might cause serious issues. Even badly corroded boards can be repaired but it's beyond most people to do so and if serious repair is needed I highly siggest sending the board to a VERY GOOD repairman/repairlady. Many people do these sorts of repair but fail to properly deal with the batteryleak. Removing all the chips and components in the area, brushing away the solder mask ot expose the copper and properly neutralizing everything is the way to go. It's a time consuming process but not strictly difficult. It's just that many just use vinagre and alcohol without removing chips, sockets and other components and thus not really dealing with the problem.
So I would say removing the Zorro-card and unplugging the HDD (taking closeup pictures of the battery area) and trying to boot straight to the kickstart-screen would be a logical next step. If issues are still there then taking the whole thing apart and giving it a proper clean with isopropol alcohol, removing all socketed ramchips and cleaning the sockets and chips with contact cleaner, and opening the PSU and giving it proper blast of compressed air would be your next steps.
And I have to say giving the paintjob a proper overhaul doesn't sound like a bad idea, and a fun project. I look forward to seeing your results. And also, welcome to the rather exclusive A3000-owners club. It's really important that we get that wonderful machine up and running
And to try to answer some of your other questions:
Getting Workbench disks is important. An alternative might be to set up a drive on a PC or getting a CF with OS preinstalled, and later making your own floppies, but I would suggest getting a copy of the operating system. And I would suggest getting Workbench 3.1.4 (with kickstart ROMS for the A3000) while your at it. It's a nice stable version and it has lots of nice quality of life improvements compared to 3.1 or 3.9 (if you didn't know, 3.9 is not the most recent version of Workbench. 3.1.4 is).
Getting a SCSI2SD-solution would be very good. SCSI2SD v6 is the fastest, but v5 is also nice. If you have the money for v6 I reccommend that one as v5 is significantly slower then the A3000's internal SCSI-controller is able to go. If you do that then you can set up your HDD on your PC (using WinUAE) before transfering the SD to your A3000. When buying AmigaOS 3.1.4 you get all the digital files needed to run a WinUAE setup that mirrors your actual amiga so it makes things easier.
The A3000 is a machine that really doesn't NEED any upgrades or additional mods to be an awesome machine, but If you want to do some fun things with it then getting an external SCSI CD-rom is nice (make sure terminators are set correctly), and perhaps an extarnal ZIP-drive (makes it easy to boot different operative systems etc.), getting a Zorro-III ethernet controller (xsurf-100 can be upgraded with USB addon, making it an USB and ethernet card). A 030 will be able to handle all amiga-related websites failry easily. Dowloading stuff from Aminet, games from WHDownload and stuff like that is so much easier then transfering by floppies or serial. Getting a 3D-printed A3000 eject button would be nice (even if you can't get the drive working, buying a new one probably won't come with the correct eject-button).