Yes, almost every game plays on the Everdrive exactly as it does on the original cartridge. And the Everdrive 64 will play games from any region, it doesn't care if the game is PAL, NTSC, or JAP. The problem is that not every TV can display all game regions, but that's down to the TV, most PAL TVs can display all regions, and any (as far as I know) NTSC TV can display JAP too, but not always PAL games.
Regarding the Everdrive 64 itself, all PAL and NTSC games work fine on the Everdrive 64, with the sole exception of Pokemon Stadium 2, which has some corrupted graphics on the version 2.0 and 2.5 of the Everdrive 64. On the version 3 and version X7 of the Everdrive 64, Pokemon Stadium 2 works fine, meaning that all NTSC and PAL N64 games will work for you.
The other region of N64 games is JAP, Japanese games. Most JAP games do work on any Everdrive 64; the exceptions are the few games that have extra hardware in their game cartridge - there are four or so of these (that I know of), such as a Tetris game that can read your heart beat (seriously - it has a little connection you attach to your chest), a Pokemon game that uses a microphone so you can speak to the game, a game that has a realtime clock built into the cartridge so it always knows the time and date, and a shooter that used a non-standard gamesave. All four of these games are JAP only, no PAL or NTSC N64 game ever had extra hardware in the game cartridge, so there are no problems with non-JAP games.
An Everdrive 64 allows you to keep all of the N64's library on one cartridge, which is extremely convenient. It also allows you to play many NES and Gameboy games (via emulators contained in the Everdrive's software), to play the 64DD games (the 64DD is an add on disc drive, that was only released in Japan), and the Aleck64 games (the Aleck64 games are arcade games that were released in Japanese arcades).
It also allows you to play game translations (such as the English translations of games that were only officially released in Japanese, such as Sin and Punishment, Wonder Project J2, Densha de Go! 64 (a Train driving game), Custom Robo, Last Legion UX , etc), and also to play most of the many game mods (which add new levels, or new weapons, etc, to many of the best N64 games, such as Goldeneye, Perfect Dark, The Legend of Zelda: Ocarina of Time, Mario Kart 64, Super Mario 64, F-Zero X, Super Smash Bros., etc).
Basically, an Everdrive transforms the N64, allowing you to play lots of games and mod that you wouldn't otherwise be able to do.