Translation:Of course I am fine.
You're right on "mochiron". Just for clarity, though, you probably wouldn't say "genki desu" to confirm you're okay after hurting your knee. The word "genki" is "fine" in the sense that you would respond to "How are you doing?". "daijoubu" would be more appropriate in the knee-hurting context.
The "u" and "i" parts are often not pronounced unless any of the romaji letter before or after it, if any, are voiced.
The pitch change also tends to influence how much the vowel ("u" or "i") is pronounced, from what I have seen. It is typically more pronounced when pitch rises (e.g. 橋 - "hashi" = bridge, rising pitch vs. 箸 - "hashi = chopsticks, falling pitch).
It's not incorrect per se to pronounce です as "desu", it'll just sound more "articulated". Same goes with e.g. "masu". And in writing, it is always spelled out.
Romaji does its best to replicate the sounds of each syllable in Japanese. す is ”su" and may be used anywhere in a word, but typically you won't pronounce the "u" part if it's at the end. In other words you will notice the syllables are sort of smashed together so that the vowel sound is absent, as in the word for ceremony, shiki. You'd say shki, but would still spell it as shiki in romaji. I encourage everyone to learn hiragana as soon as possible, and even Kanji, it makes learning much easier (less difficult?) in the long run.
In general, I would say no, it's not aggressive but rather emphatic. The person means "of course" in a sincere way. If you haven't seen someone in a long time, you might want to assure them that you've been well, so you say もちろん元気です, of course I've been well! Or using the word in a different way, you might have a youtuber whose brand is being "genki" (cheerful, lively, high energy), so they might use it as a kind of catchphrase like もちろん元気 (of course I'm cheerful like usual!)
I think it's the definition of the word itself rather than context that will help you understand better. "Fine" is a very Japanese-English translation of 元気 (genki). We have no English equivalent to this concept, but it involves being healthy, cheerful, in good spirits, well in body and soul... People are genki, objects cannot be genki.