Yes you are using that right ^^
私も元気です is correct. It would mean “I am also good” (も is an inclusive particle!)
Here, I’ll break down what I said:
Summary
私はフーヲロジストですが
“I am Whologist but”
Here, が means “but” (Like けど or でも)
フーちゃんと呼んでいいです
You can call me “Who-chan”
を - Particle for verbs (I don’t know how to explain things I’m sorry
呼んで - て-form of 呼ぶ (to call, to name)
て-form + いい means “You can” or “It’s ok to”
呼んでいいです - You can call me~
私も月が好きです!
If you want to say “that’s why ~,” the easiest way imo is to use the [reason]から[result] format. For example, 月が好きだから、「Mooni」という名前を使います。(つきがすきだから、「Mooni」というなまえをつかいます。) The だ is the shortened form of the verb です、and is necessary after adjectives and nouns when using から (I think, but it’s been so long since I learned the grammar that I do it by instinct now, so if I’m wrong please correct me!)
Literally: because I like the Moon, I use the name “Mooni.”
Have you learned this grammar before? ^^ I used another example in the first sentence I wrote.
Edit: Whologist beat me to it! Thank you for your explanation!
For context: the 笑 is the kanji for the verb わらう which means to laugh or smile, so it’s often used like “lol” in typed conversations!
もしかして、学生ですか?
Are you perhaps a student?
Also, it’s very easy to get discouraged, so don’t worry about getting things perfect! Japanese has one of the steepest language learning curves for (native) English speakers (which you may be?), so the best way to get better IS to make mistakes, ask questions, and keep practicing! I work in Japan and I make more mistakes every day than I can count, but I’ve learned so much!