読む and 書く: reading and writing
Distinguish reading content, writing location, and understanding.
What you read
読む (よむ) means reading text. 「説明を読みました」 says that the speaker read the explanation, not necessarily aloud. Reading aloud can be specified with 「声に出して読む」.
「最後まで読みました」 means reading to the end, while 「途中まで読みました」 means reading only partway. Pay attention to the expression showing how far the reading went.
Content and writing location
書く (かく) means writing characters or text. In 「紙に名前を書きました」, 紙に is the writing location and 名前を the content. The speaker wrote a name on paper rather than read the paper.
「鉛筆で書きました」 means writing with a pencil. Distinguishing に for location from で for the tool lets you explain both where and with what something was written.
Reading and understanding
「読めますが、意味は分かりません」 means being able to read it but not understanding its meaning. Reading ability and understanding differ. 「読んでから書きました」 describes reading and then writing, both completed actions.
「読まずに答えを書きました」 means writing the answer without reading. Writing was completed, but reading did not occur; preserve that scope of negation.
What does ここに indicate in 「ここに書いてください」?
Show the answer and why
It means 'here,' the place to write. The content comes from the context or an object.