Are these correct?

Zevkli bir biçemi var
(She has a tasteful style)

Ne biçim bir biçemi var!
(What kind of style does she have!)

Well, my dictionary puts 1 translation for biçem: style.

For biçim it says: shape, form, style, and then also the "kind of" meaning.

I also thought that tarz would be a synonym for "biçem" (but I didn't know it would be better to use tarz when you're clearly talking about someone's way of dressing for example.)

"Tür" would be a synonym for "biçim", I thought.

But from what you replied, it seems like "biçim" and "biçem" are the same word, just written differently?

Bottom line is, as you might have guessed: those words confuse me! :-S

No, in fact, I thought you wanted to write "biçim" but you wrote it "biçem", because I didn't know "biçem" existed as a word, but then I saw it in the turkish class dictionary and was surprised.
I also looked it up in TDK dictionary (turkish-turkish), it exists there as well, and it says "üslup" for it, üslup is a word that's used on daily basis, it's a turkish word of arabic origin, then I guess "biçem" is the turkish synonym for "üslup". Sometimes turkish synonyms exist but arabic ones are known and used more.
Anyway, "üslup" and "tarz" are quite similar in meaning, though tarz is used more for the style of clothes, decoration etc.

http://www.tdk.gov.tr/TR/SozBul.aspx?F6E10F8892433CFFAAF6AA849816B2EF05A79F75456518CA

I'd give some examples of usage, maybe it helps

Konuşma tarzı = konuşma üslubu (they're equal)
şairin üslubu/tarzı
sanatçının üslubu/tarzı
şiirin üslubu/tarzı

Below are examples where "tarz" is used nearly always
Giyim tarzı, müzik tarzı, saç tarzı


They usually appear completely equal in meaning and usage, except for hair,clothes,music "tarz" is preferred and used.
And "üslup" used alone can mean "way of expressing/talking/style of behaviour) One can say in a debate "Ãœslubunuzu hiç beğenmedim", and it means I did not like the way you talk(impolite or insulting). Here, "tarz" could only be used as "Konuşma tarzınızı beğenmedim"

Biçim is different from them, it's more about form as you said.
"Saç biçimi" (short, long etc) while "saç tarzı" is about "sporty, trendy etc"
Arabanın biçimi (round, with sharp corners, about dimensions)
Arabanın tarzı (sporty, feminine, race)

I hope these helped though there must be more to say about them.

-em suffix is not productive as -im suffix. When you hear -im suffix for the first time it sounds quite natural. You can figure out what it means easily. -em suffix is not so productive at all.

you cannot many -em derivatives.
tut-am kur-am dön-em etc.

Seems like they tried this on biç- verb. But it is not widely accepted.

Seems like biçem is more used in IT field when you google.

By the way "style" word has entered Turkish as "stil" and "biçem" is offered for it.