The 3 in 1 is what would be called an expansion pack. That means yeah you need a DS flash cart to manage it really. You can flash a single game to it and have it play and save on a GBA or GB player or something but most would instead opt for a standalone flash cart if they knew they were going to be doing that.
Yes you can use goomba to play GB/GBC games, why you would if you have a DS slot flash cart available to you I am less sure (DS emulators are better in most ways).
If you get an EZ4 (the standalone flash cart that EZFlash offer).
Space wise.
The 3 in 1 is again an expansion pack so it then becomes whatever you can fit on your DS slot flash cart (if it is SDHC then 32 gigs most likely, the whole GBA set is less than that).
The EZ4 got SDHC support a while back so up to 32 gigs there, however if you get an older miniSD version (they are still sold in some places) then I don't suggest the microSD to miniSD adapters. If you get the new microSD size or one of the others then that is fine.
Choice link
http://gbatemp.net/threads/buying-a-gba-flash-cart-in-2013.341203/page-18#post-4756995
You may have also read about 32 megabyte and 16 megabyte sections, I should explain those.
GBA games need very fast memory to run, faster than NAND based stuff like seen in SD cards. This means things get copied to fast memory first, two types were commonly available at the time and cheap enough to be viable. Those being NOR and PSRAM.
GBA games went up to 32 megabytes (list of them in that link above) so if you have one of those you need to make sure you have the space.
The 3 in 1, present EZ4, miniSD EZ4 and EZ4 lite all have 16 megabytes of PSRAM. This is fast to write but loses the data on power down (your saves are stored elsewhere by battery backed memory). They also all have 32 megabytes of accessible NOR memory. This is slow to write (several minutes) and has quirks (write files 1,2,3 and 4 and want to delete 2 then 3 and 4 are coming off as well) but stays there until you erase it.
Oh and people will speak of game sizes in megabits a lot (historical reasons, nothing special really), multiply megabytes by 8 to get it. It does also mean you will hear things like 256 section (32 megabytes of NOR remember).
There are two older models of EZ4 called the EZ4 lite compact and EZ4 lite deluxe. The lite compact lacked much in the way of GBA space as it was aimed at people wanting a cheap DS flash cart so don't get it, the lite deluxe has 32 megabytes of PSRAM and 48 of NOR. The lite deluxe can be found still (
http://www.shop01media.com/en/EZ-Flash being near you if your flag is accurate). Both are DS lite dust cover sized so smaller than GBA slot flash carts.
I should also say there is a new GBA flash cart from the everdrive people out now. It is quite a bit more expensive and larger than a GBA cart, equally I listed all the trouble games I know of in that link above and the only thing different in compatibility is pokemon and boktai games have a proper real time clock rather than a software driven one. Still it is an option that is available to you and everdrive do make good carts, with some of the features there being things I would be happy to see on the EZ4.