Welcome! I just got into arcade stuff two years ago. Before I answer your questions, I want to answer some questions about the arcade hobby that you didn't ask.
Do I need to know how to do DIY electronics work?
For CPS3, you could just buy a working setup. However, as I got more platforms, I soon found I couldn't get far without getting tools and learning to DIY for a few things (crimping cables, soldering, programming EPROMs) since a lot of the tasks seemed like a waste to outsource (e.g. ship heavy, fragile hardware to solder 2 wires, or fix a kick harness that was shipped with a few wires misrouted).
Do I need original hardware to have the arcade-perfect experience?
For CPS3, yes,
but I'm hopeful that Jotego's long-term CPS3 implementation for MiSTer will be just as good as his CPS2 implementation-- which is to say, indistinguishable to players at Evo. MiSTer isn't cheap, either, but it's a lot cheaper than getting a CPS3 Ultra SIMMs setup.
(Important caveat: Most MiSTer cores aren't cycle-accurate to original hardware, and are arguably comparable to software emulation with less input lag.)
As for the questions you did ask:
Battery:
CPS3 doesn't need a mainboard battery for any reason. If your motherboard came with one, you can remove it.
CD vs. drive emulator etc.: Buy a working setup. If it does what you want, you're done! Modding a stock CPS3 just to run a single game from alternate media sounds like a waste of time and money to me. But if you want easy access to all SF3 revs and the three other comparatively obscure games on CPS3, you could get
Darksoft Ultra CPS3 SIMMs with the Ultra BIOS Cart.
Recap: IMO don't bother unless A) there's a cap clearly behaving badly, B) there's a cap known to be troublesome on other instances of that board (and I'm not aware of any on the CPS3 except the battery), or C) you're sending it to a modder who's doing some other install anyway. If you don't know what you're doing, you can make things worse. I have done some preventative shotgun recaps like what you're describing, but my last one broke my System 16B and it's hard to justify spending a lovely day investigating a non-urgent problem.
Connecting your stick/controller to a supergun: You'll need a USB-to-DB15 adapter. I like
USB2DB15, but I've also used the (rarely available) Undamned USB adapter with Brook adapters.
Connecting to a TV: You need a supergun, kick harness, power supply, A/V cable, and (if connecting to a modern display) an upscaler or
CPS3 Digital AV mod. I have something to say about each of these, but it's getting late. My recommendation is to try to get a
Home Arcade System supergun, but RGB just sold out of the latest batch and I'm not sure when the next is due.
How much should I expect to spend? This stuff adds up quickly. $1,000 for just Third Strike or $1,300+ with a Darksoft multi, $130 supergun + $10 CPS2/CPS3 kick harness, $35 power supply, $40 A/V cable, $75-750 for a good upscaler to a modern display (or alternatively, get a CRT), 2x $69 USB2DB15 adapters...
