If the map is acting as a game board then my point about removing the grid doesn't really apply. I'd still suggest dropping that odd looking warping effect though as it really doesn't make sense.
Well, the "grids" on real maps are a way of representing a coordinate system and are called "graticules". Most commonly the coordinate system is latitude and longitude, but sometimes another one like UTM is used. Modern street maps use a locator grid, often with alphanumeric coordinates but that's a fairly specific thing.
If you're using it for a game, then I'd think the mechanics of the game would dictate the grid used. If you want a real life lat-lon graticule, then it actually depends on where the map is, and what it's for. For a reasonable lat-lon graticule for a small area, you can use this tutorial to come up with something reasonable.
They are your maps so you can do as you wish. I was just giving my opinion. I think outright crumpling looks off as a hand drawn map is something that is fairly valuable and is going to be taken care of. Texture from the paper (or whatever it's drawn on) is a fine idea, but you might want to think about what that texture implies about the map and pick the right one. Again it's your map so if you like the crumpled look anyway, go with it.
Well, think about how the rest of the map was drawn, and how that might be applied to drawing a border. A simple line or double line works just fine, or you can apply a simple drawn pattern between two lines. Checked borders are common and usually act as a supplement to the graticule (which means they need to line up) You can also make them strictly decorative, but it doesn't look very good in combination with a grid unless they are aligned.
Here's one of my maps that has a checked border aligned with a graticule to demonstrate: