Oop. '3. How will I get all the details of the face when I'm at a distance from them?'
You don't NEED to. You only need all the details if you want your art to be a SPECIFIC person. And even then, again, only if you want it to be a specific person and you want everybody to notice every little detail AND read it as 'important'.
As an example, I'll use myself, since I'm a fairly distinctive person. Someone wants to represent me, all they have to do is draw a skinny chick with pink hair and that is going to be me. Just like the tall guy wearing blowing glasses is one brother, the tall guy in a lab coat and glasses is the other, and the one with the mustache and ponytail is my dad.
I could draw little mushy people made of circles and triangles, slap those details on them and put it on the wall, and when the studio is full of customers during a show or whatever they would ALL be able to pick us out- because we showed them the relevant details.
Honey, they are YOUR characters, you're obviously not going for portrait or photo realistic work. You just need to mix it up some. And you are worrying WAY to much! You are primarily anime-style, yes? That's all about simplifying things, just like any other type of cartoon. When was the last time you looked at animation and thought the people actually looked exactly like they should? Animation, cartooning, all that stuff relies on shorthand. That is why the smart character wears glasses and the wild one has pink hair. ONE glance needs to tell you everything you need to know about the character (to enjoy it). You don't have time or the NEED to show every little detail.
And people will fill that in if you don't show it to them, anyway. When you draw, you show people what you WANT THEM TO NOTICE, and let them fill in the rest.
Think about fingernails in cartoons. We know people have fingernails. But how many cartoons do you see where they don't? Why? Because drawing them on there means 'Hey! Look at this! This detail is important!' If you took a cartoon character and converted them in your head to a 'real' person, they would have fingernails, right? They aren't some freaky nail-less critter.
The point is, you don't have to draw everything. You just have to put enough in there so our brain goes 'Ah!' and puts the right details in there. Eye shape, nose shape, skin color, hair color and style, dress. That's all you need.
The more detail you put in there, the less the persons brain fills in. This might seem like a good thing- but you have to realize that, since details mean 'notice this', if your details are WRONG it makes your drawing read wrong- or rather, it makes it read with that as important.
Draw a stick person, and we'll assume it's a generic, healthy human standing there. It's a placeholder for 'person'. If you are asked to imagine details, they would look 'average'.
Now take a very, very detailed painting with some anatomical errors. Arms are too small, face is misshapen, etc. What does it look like? It looks like someone who is deformed. A very detailed hand with horrible anatomy looks like a really deformed hand. The artist could be trying to make it look like an image of the most perfect person on the planet, but that doesn't matter. We have been told to notice things, and what we have been told to notice is abnormal. So the person we see is deformed.
1. How will I know what race the person is?
I dunno, how WILL you know? You're the one doing the drawing. Certain features are common to certain ethnic groups and/or areas. All you have to do is find the most common of these and use them. For an easy example, think about eye shape. There is a distinct difference in people from Asia and from Europe- even if you don't get down to the nitpicky bits that place a person in a specific region (eg. Chinese vs Japanese eyes).
2. What if the person is of mixed race?
YOU are making this person. Pick the details you want and stick them on there. Look at a person of one race. Look at a person of another race. Go 'Hm, I like this nose, these eyes' etc.
3. How will I get all the details of the face when I'm at a distance from them?
You may find it easier to draw from photos or from people you know who will sit still for you. It will be hard to get close enough to people to draw them properly without them noticing. Sneaky photos are fine, you're just doing it for drawing practice.
As for race, don't worry about it, just draw them. When you want to draw a specific character of a certain race, then you can look up people of that race on the internet.
Their country of origin or genetic background is completely irrelevant. You don't have to worry about avoiding stereotypes, or even what ethnicity a model is, if you just DRAW WHAT YOU SEE. Observation is more important than expectation. So draw what you see, not what you think you see, then you'll be golden.
You want to take a picture of someone? Go for it. Might want to be sneaky about it or ask permission first, but do what you want. Just remember, the point of drawing from life is not necessarily to produce polished, finished work, but to study form. It'll also teach you to draw more quickly.
Well, seeing as you're getting it wrong *right now* I don't really see why you are so worried about getting it wrong in the future. You're already there. Whatever horrible thing was going to happen is happening already. If you think putting slightly the wrong nose on a recognizeably ethnic face might be wrong, how much more wrong is it to give every person generic cartoon features and overlook the diversity of faces altogether?
Basically, every "problem" that you have brought up here is not a problem, it's a delaying tactic for you to not have to try something new. If you actually wanted to do anything but draw the same beefy girl forever, you would just go out and try this and solve the problems as they appeared. But some people told you that you should do this, so now you're casting about for reasons not to.
For one and two, check out `majnouna She's got some AMAZING tutorials about how to differentiate race in sketches.
For three, don't start off with distance sketches. Start out drawing people up close in places like a restaurant, library, or other public area. I don't advise drawing people you know well though, not right off the bat, because you'll end up drawing what you think you see instead of what you actually see.
I think you are over-thinking it a bit. Don't worry about being able to identify the race of every person you're drawing, you can worry about applying a recognizable race to your characters later. No matter what race a person is, they all have the same general features. Just worry about being able to realistically capture those features, then worry about race later.