No, you can set mame to maintain the original aspect ratio of the game. There will be space 'left over' on the sides; however one fun thing you can do with many games is have mame overlay 'art' from the original cabinet to fill out that space, such as the original bezel. There are examples of this in various projects.
The 'leaving the display exposed' that you're talking about is usually called a 'showcase' configuration. There are several examples of that around also.
When it comes to resolution in LCDs, at that size you're going to see essentially two options, 1080p (or close to it) and 720p. Either one provides enough resolution to display all of the pixels of the game, but since you've already gone the LCD route, it makes sense to go ahead and get a higher res display in case you ever decide to play any other games on it which might benefit from a higher resolution. Higher resolutions are also beneficial when using the various 'filters' available in mame to 'smooth' the look of the graphics, or add scanline effects. This is up to personal preference if you like the look of them.
Vector games also benefit from a higher resolution display.