It's hard to give answers to these questions that are both simple and true, because phonetics is a complicated topic.
1. /ə/ and /ʌ/
You don't need to worry too much about the distinction between /ə/ and /ʌ/. The two vowels rarely contrast; in most varieties of American English, they actually never contrast. It's hard for even native English speakers to hear a difference. The schwa /ə/ is definitely not like a short "e" sound. The symbol /ə/ can represent slightly different sounds depending on the surrounding sounds, but in general it will be understandable if you pronounce it like the "u" in "up."
2. English monophthongs and their transcription
Different varieties of English have different sets of vowels. Also, there are different ways you can transcribe pronunciation depending on what you're focusing on. John Wells describes some systems in the linked article. The vowels you list are the traditional symbols for vowel monophthongs used in British IPA phonemic transcription.
You'll notice that this system:
ʌ ("up"), ɑː ("mark"), æ ("trap"), ɛ (e) ("bet"), ə ("the"), ɜː ("nurse"),
ɪ ("kit"), iː ("fleece"), ɒ ("lot"), ɔː ("thought"), ʊ ("book"), uː ("boot")
is redundant. The long vowels all have different symbols from the short vowels. So it's possible to leave out the length markers:
ʌ ("up"), ɑ ("mark"), æ ("trap"), ɛ (e) ("bet"), ə ("the"), ɜ ("nurse"),
ɪ ("kit"), i ("fleece"), ɒ ("lot"), ɔ ("thought"), ʊ ("book"), u ("boot")
Or you can keep the length markers, but use fewer vowel symbols:
ʌ ("up"), aː ("mark"), a ("trap"), e ("bet"), ə ("the"), əː ("nurse"),
i ("kit"), iː ("fleece"), ɔ ("lot"), ɔː ("thought"), u ("book"), uː ("boot")
(I have not found many resources that actually use a purely "qualitative" system like this. One example though is the book Making New Words: Morphological Derivation in English by R. M. W Dixon.)
English also has a rich inventory of diphthongal vowel sounds (again, the precise set is different for different accents). These are written in the IPA with two adjacent vowel letters, but they are really treated as single sounds by English speakers, so you should learn how to pronounce them as units.
British English lacks consonantal "r" at the end of syllables or r-coloring on vowels. In older varieties of British English, sequences of a vowel followed by "r" were often pronounced as "centering diphthongs" ending in a non-syllabic schwa sound: for example, "square" would be transcribed /skwɛə/ or /skweə/, "near" would be transcribed /nɪə/, and "cure" would be transcribed /kjʊə/. However, in modern British English pronunciation these "centering diphthongs" are often replaced in pronunciation by long vowels, some of which are of a different quality from the other pre-existing long vowels: "square" /skwɛː/, "near" /nɪː/, "cure" /kjɔː/.
3. How to tell if a vowel is long
If you have an IPA transcription, this is fairly easy: long vowels will generally have a length marker after them, as in /ɑːm/ and /tɜːn/.
If you just have the spelling in English, it is often difficult or impossible to figure out how to pronounce a word. Long vowels may be written in many ways.
4. Accents
It really depends on what you want to do, and what features are part of your accent. Some accents are distracting or difficult to understand, while others are barely noticeable. Your pronunciation of the first vowel in "professional" doesn't seem like it will impede communication, in my opinion.
5. CVC
I don't know exactly what your friend means, but I don't think this is a useful way for you to think about the structure of English words. As you say, there are many words of the structure /CVː/. There are also words that start with vowels (/VC/, /Vː/) or that start or end with more than one consonant (/CCVː/, /CCVC/, etc.).
Best Answer
The answer is that it depends on what purpose you have in assigning it, or what set of rules you are following.
From the point of view of phonetics, the first thing to realise is that letters are not vowels or consonants: they represent sounds which may be vowels or consonants (and in the case of "y" possibly both).
The next point is that bifurcation into vowels and consonants is too simple: phoneticians recognise other possibilities such as "semivowel" — which "y" often is.
It is clear that in "Yvonne" and "mystery" all the "y"s represent vowel sounds.
I would say that in "yacht" and "Yeltsin" they represent semivowels (which you can call consonants if you like.)
I would disagree strongly with decoz.com (quoted in Mehper's answer) about "Kay" and "Sydney" — I think it is preposterous to say that "y" is representing a consonant in those. In the case of "Sydney", it is part of a way of writing a simple vowel sound; in "Kay" it is part of a way of writing a long vowel sound or a diphthong, depending on dialect. (A diphthong consists of two vowels or a vowel and a semivowel depending on how you want to analyse it).