The Duolingo course teaches that و = uu, ا = aa & so on.
(I made it big so the arabic would be clearer.)
My question is that is it actually true? Does Arabic not have sounds like o, e, but instead they say it a bit differently (e->aa, uu->o)?
Yes. Arabic has 3 main vowels only and they can be either short (shown by using diacritics or mark on and below letter) or long (shown using letters).
In dialects nowadays, you might find other vowels we hear in European languages and others, like E and O, but originally, in Arabic, those do not exist. It's either A, I or U.
Well, not exactly. Sometimes, there are words that have been fixed in pronunciation via the Arabic language assemblies across the Arab word. e.g. "Electron" would إلكترون and "Proton" would be بروتون.
I remember when I was a kid and used to buy language teaching books (small ones like those for travelers), the author, typically an Arab, would make his own combination of signs and diacritics to approximate the pronunciations for vowels and consonants to the Arab reader, and even borrowing some letters from other languages like Farsi for example. I remember one author suggested ـاِ (that is long vowel A with Kasrah below it, which does not make sense in Arabic) - just to explain a long "E" or "AI" sound, as in "main".
As for proper names, like Peter or Michael for example, there had been also some conventions on how to write them (without the need to use diacritics).
Not really. Not much confusion there. Because in our dialect we do (or we have developed) such vowels already like O and E (and in some instances they do make a difference in the meaning). In fact, the vowels in the dialects are also one of the features that distinguishes one dialect from another even though the same word is spoken. e.g. "Kuwait" (which in standard Arabic is Kuwayt) is Kwait in our dialect. Egyptians would say it Kuweet. In Arabic, they are written the same كويت it is just how it sounds in the dialect that is different.
In social media or text msging we do type our dialect but there is no such thing as proper dialect orthography or writing system. In fact, we mostly use the standard Arabic words because it is what we say except for some technical words maybe that are from english ... then we do substitute letters... and despite the absence of an orthographic rule, it is common among us to type certain words in certain way.
We use something called (حركات-harakat ) and we also use something called ( tanwen-تنوين) the harakat are : ( A َ ) this one is fatha and it makes the letter have an A sound for example( سَ -sa) and its always written on top of the letter. (E ِ ) this one is kasra and it has an E sound and it always written below the letter for example (سِ-se or just saying the letter c ). (U ُ ) This one is damma and it has a U sound (سُ-SU). ( ْ )for this one i am not sure how to explain it but its very similar to the sound the we use when ending a word that has no vowels, its like saying : in or is (سْ). for the tanwen you just double the haraka and make two of them for example like : ( ً ٍ ٌ ) they sound like ( AN اً -EN اٍ - ON اٌ ) and they are mostly used at the end of words. and there is also the shada which looks like this( ّ ) and when this is used on a letter its almost like doubling it and when you are saying the letter you should hold it for a bit. and there is also the hamzat AlMaad ( آ ) when you see this one that means you have to extend the sound like saying AAAA. I hope this helps.