Emulator only patches are mainly a problem for the NES and SNES, possibly some arcade stuff as well but I am not sure there. The N64 will see things like high resolution textures which is not going to happen on hardware, though it too has a few hacks that play fast and loose with the hardware specs, and there is a similar story for the GC and Wii on the texture front. Similarly many emulators will have Lua support (lua is scripting language not unlike python, perl and such) and you can get scripts that are only going to work on lua supporting emulators but that is not usually a problem for those wanting to do ROM hacks as they traditionally appear.
The megadrive/genesis tends not to have this and I have yet to see stuff like that for the GBA and newer handhelds.
On the DS stuff. DS flash carts are probably the biggest of any flash cart scene and it is mostly over/sorted. If you are playing on your relatively current/unhacked 3ds then you do not have many options. Said options are either a DSTwo or whichever of the R4i golds are doing good this month.
The DSTwo is quite expensive at about $40 USD, though compared to the everdrive line... it might drop in the next few weeks as the DSTwo+ is coming out (they came back for the 3ds and that is their first effort) but do not expect much.
It is a bit of a battery hog (though not enough to really hurt it) as it has an onboard processor that allows it to emulate GBA, SNES (native SNES emulation on the DS is nice but not as good as a DSTwo) and play videos and do all sorts of things. On the DS side of things it is an exceptional flash cart, arguably the best that the DS saw in terms of features (it has pretty good savestates, a nice cheat engine, the ability to search for cheats during play, an in game guide and more besides). The GBA emulation is not perfect, especially compared to native hardware flash carts as will be discussed in a little bit, but it will allow you to play most of the GBA library and most hacks should work just fine if the original game does.
The R4i gold should support the wood firmware (a third party firmware that was pretty nice when all was said and done), should support just about every DS game going, will have cheats and some other nice features but no onboard processor to help emulation. It should also bypass the 3ds anti DS flash cart checks, Nintendo could update said checks again next firmware update and stop it for the 3ds but they seem to have given up and are focusing on 3ds mode hacks instead.
Alas I am going to have to either look it up and report back or get you to look it up -- the R4 was the first popular DS slot flash cart (it was drag and drop and quite cheap) so R4 became the word used by those not in the know to refer to every DS flash cart (think how some people use jailbreak instead of hack). Consequently there were hundreds of flash carts with R4 somewhere in the name and most of them were junk made by fly by night (even by borderline illegal Chinese hacking device maker standards) companies.
No DS flash cart I know of has onboard soft patching if that is what you were heading towards, especially as the IPS format pretty much completely fails to do anything useful on the DS so a whole bunch of quite complex formats got used. With that said most DS ROM hacks came in patch form though and will spit out a nice .nds file that you just copy to your microSD like any other ROM so it is not too troublesome.
Back to the 3 in 1 stuff.
The DS and DS lite both have a slot for GBA games (the DSi and 3ds do not) so if you have one of those and a DS flash cart you can happily use that to manage the GBA slot and GBA flash cart. There are not really any DS mode GBA emulators outside of so called enhanced flash carts, which for most means just the Supercard DSTwo, so you will have to use GBA mode and that needs something in the GBA slot.
Quick version here is the GBA needs very fast memory to run, far faster than NAND memory like that of CF or SD cards and especially those at the time, so that means either NOR memory or some kind of RAM (PSRAM mainly) if you wanted to make an affordable GBA flash cart. The 3 in 1 has NOR and PSRAM (32 megabytes and 16 megabytes respectively, recall that GBA games top out at 256mbit/32 megabytes (divide by 8 if you are not used to it) but most were 128mbit/16 megabytes or less), however it has no onboard storage or external storage so it needs a DS slot device to manage it. Not having this onboard storage or need for self management makes it a bit cheaper than the EZ4 which does run standalone and does have external storage. The new microSD GBA sized EZ4 will be about $50 USD and the 3 in 1 will be around half that for a DS lite dust cover sized one, good luck finding a GBA sized one for a sensible price these days, though if you want you can get the sandpaper out and have some fun it will fit and run in a regular sized GBA slot.
Oh and for the wii softmodding then https://sites.google.com/site/completesg/
is the usual link. If you know what you are doing there are things you can skip and tweak during that but for that will not send you wrong. Wii emulators are pretty good and if you can get a GC controller or a classic controller you can have a great time.
Otherwise... yeah most PC graphics cards and laptops have VGA or DVI/HDMI out which plugs directly into the back of most modern TVs with a cheap and cheerful cable, and in the case of HDMI even does sound as well. The hardest part will tend to come if the HDMI port has overscan (the picture is too big or too small for your screen) that you need to fiddle with, and a search for AMD/ATI, nvidia or intel overscan on windows XP, vista, 7, 8, 10, linux (delete as appropriate) will get you most places you need to go for that one -- typically though if there is an advanced menu where you change your resolution it will be in that. You may have to enable game mode, PC/computer mode or something on your TV too to get the latency down but that should be an option mentioned in the manual. There are any number of controller adapters or nice controllers that plug into USB or go over bluetooth you can use if a keyboard is not your preferred way to play old games. I have a USB 360 controller that I used for most things, if you have wii stuff then a simple bluetooth dongle if your machine does not have it will do there (PS3 will also work over bluetooth, sadly no 360 wireless in a sensible manner).
DS games might be harder to emulate on a really slow PC but basically anything will emulate the GBA and 16 bit era and backwards.