No, you can't. You cannot place が directly after a verb.

If you MUST use が for some reason, you can put it after the か.　

「かが」 is correct and natural.

何かがおかしい。
誰かがボクのお金を持っていった。
何か食べたい、でも何を食べたいかが分からない。

In that case, only 知る can be used (yet in the past tense & negative form).

Person B most likely didn't expect Person A to know the mushroom's name. Accordingly, Person A could not sound indifferent by using a form of 知る.

本当にありがとうございました、masaegu さん。:)

Is べつに considered curt in all situations, or just when your whole response is べつに?

Mostly when that is your whole response because you can sound like you don't want to volunteer any information regarding the topic.

When you add words to べつに, it　won't sound curt.
e.g. べつにないよ、べつになんでもありません、べつにだい じょうぶだよ、べつにそういう訳ではないんだけど・・ ・, etc.

And for some similar reason you mostly 知る people and places, not 分かる, right? Say,
A: いつから彼女を知ていますか。
B：1990年からです。

but could I say something like this?
ア：性格は悪いのであの人を分かれません。

And then you shouldn't use 知る when you mean to express sympathy, right? If someone was telling me how much they missed their dead dog, then I could maybe use 分かる if I clarified it somehow ("分かります。I miss my dog too/ your dog was awesome"), but 知る would be that indifferent response?

Correct. 知て　＞　知って

Yes but there are two mistakes.
性格は　＞　性格が　There is no choice here as it's in the subordinate clause.　
分かれません　＞　分かりえません　ＯＲ　分かることができません

In that case, 知る would not sound indifferent but it would sound plain strange and inappropriate..

Ok :) Thank you Masaegu; this is the kind of thing that they drop on you in class and then ~never~ teach you the difference or exactly how to use it.

Hi, I would like to know what is the English translation for the sentences above and why "知りませんでした" and not "知りません"?

B: Can you see the thing up on that tree? That is a mushroom named "Forest Bird". They are delicious.

A: Is that so? I didn't know (about the mushrom).

I think you can see now why it's in the past tense.