Do I have to make a map of areas outside of my dungeons? For instance, a village or forest? Does the context of the Dungeon have to be mapped?
In the past I've used area maps, rather than concrete building-by-building maps. In this way I create zones within the city, so if the group needs to go to see a shopkeeper, they'll find him in the shop zone, if they need to talk to a snooty rich guy, he'll be in the snooty rich guy zone. This way, the settlement is abstract enough that if I suddenly need a candlestick maker then I know where he is, and I don't have to evict someone from a building I've already given to the butcher. It also is handy for approximate walk times and the like.
No. You may not need a map or you may be able to improvise a map.
I don't even map dungeons in advance. I found that my dungeons were a series of fights, traps, and puzzles. The maps connecting them were arbitrary when I set them up during prep, but at game time I was always stressing out about connecting the hallways correctly. Instead I just make a list of the dungeon features and improvise the order and path between those features. At most I'll have a layout for a single room.
But that doesn't mean that you shouldn't map. Try it and see if it helps your game. Learning to GM is all about learning what aspects you can improvise and what aspects need to be prepared. While I can improvise a dungeon map I can't improvise a description. Scenery has to be planned ahead of time when I take notes. With time you'll learn what you need to prep.
There is never a requirement to map out anything. I've run entire campaigns without any maps, including no dungeon maps, other than the ones in my head. It's a lot more work past a certain point, but it can be done.
Make only what maps you need; most villages can be nothing more than a dot on the regional map and be quite happily used.
If you need to map out villages, keep in mind the basic nature of a medieval hamlet or village: A central commons, houses and businesses surrounding it. Two buildings dominate a village square: The church and the tavern. Anything else is either on the square or on it's own lane. So, typically, a village might have 2-3 businesses, and they'll usually be on the square. A clockwise listing is almost as good as a map most of the time.
Maps are fun. Make maps when you want to. You don't have to map anything, ever. But you'll want to, because maps are fun.
Maps are just another tool that you have as DM to convey information to the players. When you want to convey something that is best done spatially, a map is useful.
Personally, I find I often sketch very rough maps all the time during play to give players an idea of the layout of complicated scenes, especially fights. (I only used a battle mat for AD&D once, and it turned out not to be a useful idea.) These maps generally look terrible, but they serve the purpose of the moment: to get everyone clear on what's happening and who's where.
I also found that village maps that I spent time to create outside of the game were the least interesting thing I could show my players, making the time invested have a particularly bad payoff. I don't bother with detailed village maps anymore, preferring to spend the time detailing the geography around the village (but only if it's interesting and not just "lots of forest with a road going through it").
I always draw a regional scale map (like this one) if I draw any maps at all. I find this has a good time-to-payoff ratio because it gives the players a good high-level understanding of the area where the game will take place in early sessions. Also, I just like maps.
Maps don't have to be fancy, just useful. Consider the simplicity of this map, or the sparseness of detail in this one. Each still manages to convey a significant amount of information even if they're not works of art and even if they're not packed with every possible detail. Even a small, simple map sketched in three minutes can help a lot in planning a game because it gives you insights you might not otherwise have had.
But no, you don't have to draw maps of outside areas or villages. Only if you want to. You can easily get by without them just by describing the pertinent information as your players need it during the game. But they're fun.
Map anywhere where a player needs to make a decision based on geographical layout. If the characters are going to get up to a lot of mischief there then draw a map. Doing mischief will often mean deciding which roads to use, where to put a safe house, whether the guards can see the taven from the castle etc.
Edit: If you're going to have a fight, you'll probably need a map to show where everyone is standing. If the geography matters to the fight, then draw that in too.
If it is a place they are passing through then a description of places to go there is enough.
You don't have to map everything in one go. Have a description and if the players keep returning to a place then map it. You don't have to map somewhere in great detail at first either, you can do a rough map and then fill it out in the future.
Yes and No. And it Depends.
You generally want a large overhead view of the village or forest to get a general sense of scale or extremely unique points of interest (like a waterfall, or mountain top.)
You will also want maps of "key battle points" or other points of interest. But by no means do you have to know who lives at 12 main street, or how many trees are within a 20 ft square area 70 feet into the forest from the north.
However,you do need enough information to be able to figure out that information on the fly.
Commonly in D&D players might be making their own maps, in which case you can give them vague descriptions and let them map as they please.
It all depends.