Learn German: Ich mache das selbst. - I am doing that myself.

In English, "myself" does two different jobs. It can be a reflexive pronoun receiving an action (like "I wash myself" -> Ich wasche mich). But it can also emphasize that you did something "on your own" or "personally" (like "I am doing that myself"). When it means "on my own," German uses the word selbst rather than a reflexive pronoun.

Will selbst change its spelling to match different people, like mich and dich do?

No, it stays exactly the same! While reflexive pronouns change depending on who is doing the action (ich wasche mich, du wäschst dich), selbst is an unchanging word. "You do that yourself" would just be Du machst das selbst, and "she does that herself" is Sie macht das selbst.

I sometimes hear people say selber instead of selbst. Could I use that here?

Yes, you can absolutely say Ich mache das selber. selbst and selber mean exactly the same thing in this context. selber is just a bit more casual and is mostly used in everyday spoken German, while selbst is standard for both speaking and writing.

Why does mache translate to "am doing" instead of just "do"?

German doesn't have a separate "continuous" tense (like "am doing," "is walking," or "are eating"). The simple present tense Ich mache covers both "I do" and "I am doing." You just rely on the context of the conversation to know which translation makes more sense in English.