All of your problems can be solved with one thing! Yay for simplicity!
Kanji have (generally) two different readings, the on’yomi and kun’yomi. You don’t really have to remember which is which, but using one or the other changes based on whether the kanji is on its own as a word or part of another word. 今 has both the いま reading when alone and こん when in compounds, like 心 is こころ when it’s just the word ‘heart’ but しん within a compound.
And the meaning of the kanji 半 is half, but when you actually use it as a word, it becomes 半分（はんぶん）.
Regarding your first question, kanji have different readings known as the onyomi (sound reading taken from the pronunciation of the Chinese characters) and the kunyomi (meaning reading taken from the Japanese word represented by the Chinese character) readings. They’re a fundamental part of learning kanji, and it’s not uncommon for kanji to have multiple readings of both types. There are a few general rules (usually compound words use the onyomi reading, for example), but you’ll need to pay special attention to the correct reading. WaniKani has an article on this topic found here: https://knowledge.wanikani.com/wanikani/japanese/onyomi-kunyomi/.
Your third question is related to the first (しん and がい are onyomi readings while こころ and そと are kunyomi readings). When you input the wrong reading, WaniKani will nudge you in the right direction with a gentle reminder.
As for your second question, while those words mean the same thing, they technically have different functions, but I don’t sufficiently understand the distinction myself. 半 is a noun, noun suffix, and noun prefix, while 半分 is a noun and an adverbial noun.
EDIT: You should run into 生 soon, as it is a Level 3 Kanji, and it’s an excellent example because it has a number of onyomi (せい, しょう) and kunyomi readings (い・きる, う・む, は・える, and なま [the interpunct separates the kanji’s pronunciation from its accompanying hiragana]).
Each of the kunyomi readings has a different meaning associated with “life” (“to live”, “to give birth”, “to grow”, and “live” [as in, “live bait”], respectively), while the onyomi readings are used in compound words like 人生 (じんせい, “one’s life”) and 一生 (いっしょう, “lifelong”).
The reason is because kanji by themselves are not necessarily words. Sometimes they can be words, like in your example, but not always. The readings for kanji are how you would pronounce them when they make up other words made up of multiple kanji.
I’m not sure on this one
The reason for this is because こころ is one of the readings for the kanji 心, but not the reading wanikani is looking for, so it shakes. If you try to put しん for the reading of the word 心 it will mark you wrong because that is not how that word is pronounced. To sum it up, vocabulary words only have one way to say them, while kanji can have multiple ways they are pronounced.
You’ve already got some good replies on on’yomi and kun’yomi so I won’t go over that again. Instead, I’ll try to clarify each of your questions based on that knowledge.
今 is pronounced いま when it’s used as vocabulary because it stands for the word いま. 今 has the reading of こん when it’s used in certain words such as 今夜, こんや, this evening.
半分 as a word means half of something. 半 by itself is usually used with something else to denote half of it: month-and-a-half, half an egg, half time.
こころ and しん are both actual words in Japanese that mean heart. WK shakes to let you know that it wanted the other reading. The Kanji 外 stands for the Japanese word そと. がい as a word on its own means “damage” and is represented by 害. That’s why it marks you wrong for answering がい instead of そと. Edited: See @Leebo’s clarification below.
You’re kind of conflating a few things here. The 心 vocab item will not shake if you try to answer with しん. It’s true that that’s a word, but that isn’t the word being taught here. The fact that がい can be 害 is not why がい is unacceptable for 外. がい for 外 could be its own vocab, but it would be a different item than そと.