Glaubst du, dass einer der Ansaätz iste besser ist als der anderer?

Ich glaube, man muss auf beides achten, Logik und Gefühle beide achten.

Oder sondern was Beziehungen betrifft?

"sondern" passt hier nicht rein. "Es ist nicht das Beste was Geld betrifft, sondern was Beziehungen angeht/betrifft." ist ein Satz wo man es verwenden kann. Ich finde allgemein klingt "sondern" in Fragesätzen etwas komisch.

Normalerweise sind sich mein Herz und mein Kopf einig, aber mein Willen ist schwach.

Zweites "mein" nicht zwingend notwendig klingt aber besser.

In Beziehungen verlass ich mich eher auf meine Gefühle, in aller Wahrscheinlichkeit weil die Gefühle sich einfach nicht bewältigunterdrücken lassen.

Ich folge dien Wünschen meines Herzens soweit wie möglich, weil ich weiß, dass das Vernachlässigen des Herzens meinerseits zur Unzufriedenheit führt.

Ab und zu erwünscht das Herz sich was Unrealistisches oder Falsches und in diesen Fällen muss ich bloß so hart wie möglich dasem Begehren/der Versuchung widerstehen.

"erwünscht" klingt ein bisschen altmodisch/barock, hab es im Lauf meines Lebens nur extremst selten gehört.

Mann muss immer wieder versuchen, Logik und Gefühle soweit wie es geht zu versöhnen.

Denn ohne das Achten auf Gefühle leidet mann unter unerfüllten Träumen und ohne das Achten auf Logik gerät man in Schwierigkeiten.

Glaubst du, einer Ansatz Herangehensweise ist besser als dier anderer?

I think you wanted to use "approach" here as in "one approach is better than the other". While "Ansatz" is not necessary wrong, "Herangehensweise" is a better word to express what you mean.
On the other hand, if you used "Ansatz" you would have needed to write the sentence like that: "ein Ansatz ist besser als der andere". Notice the article has changed. As "Ansatz" is masculine and it is not defined (meaning we don't know how many, what kind, etc.), We'll use "ein". The ending for "anderer" has changed as well, leaving out the "r", because it is singular and adjusts itself to the correlating subject ("Ansatz"/"Herangehensweise").

Warum, beziehungsweise warum nicht?

We have two separate questions here and in German, we put a comma everywhere we can. It looks nice, it indicates a break and separates different sentences. I'm not sure myself, but I would say that since there are two possible questions, we have two possible sentence and if you would read it out loud, there would be a tiny break between the first "warum" and "beziehungsweise".

Ich glaube, man muss auf beides, Logik und Gefühle beide, achten.

The equivalent in English: "you have to pay attention to both logic and emotion". "Both" is a weird word to translate, because it needs to be adjusted to the sentence in German. "Beide" is not a wrong translation, but it cannot be used just like that. First the placement is off. It comes before the two words it's supposed to represent. And in German, again with the comma, we want to show what "beides" represents but can't just simply put it there. It's like a tiny sentence that squeezes itself in-between another, which is why it is put in between commas.
An easy way to try out whether or not a comma might be needed is to read the sentence out loud and try noticing a break or change of voice OR you try to form the sentence without a certain chain of words. In this case "Logic und Gefühle" can easily be left out, the sentence would be just fine without it. We just wouldn't know what "beiefes" refers to. Or maybe it is also stated somewhere else in the text.

Das Beste auf kurze oder lange SAussicht?

The sentence itself sounds a bit weird. I would probably more likely say "Das Beste näher oder weiter Zukunft", mainly because I find it sounds smoother, but your sentence is just fine. I would simply use "Aussicht" instead of just "Sicht", as "Aussicht" refers more to something in the future, whether far or not.

Oder sondehern was Beziehungen betrifft?

This one is weird to explain. Technically both words have similar meaning, but "eher" is the honest equivalent of "rather" and is more fitting in this sentence.

Normalerweise sind sich mein Herz und Kopf einig, aber mein Willen ist schwach.

It's a weird sentence and probably sounded a bit different in English. I will simply correct it as it is.
"Sich" is not easy to explain... It's one of those things you simply have to get used to by listening or reading. What I can tell you is that it has to be put before the word(s) it refers to (Herz und Kopf), it's a "Pronomen" so it's always in use with a correlating subject and is therefore adjusted (mich, dich, sich,...).
"Wille" is normally written without an n. At least in this case as it is singular.

Ich weiß, was ich tun soll/muss, aber irgendwie fällt es mir schwer, mich selbst zu beherrschen.

Again, a weird sentence and not in tune with the other ones. I will again focus on what I have.
"Fällts" is not actually a word, but that mistake is made by a lot of Native Speakers as well, so don't worry. It's just easy to write, because the correct form would actually go like this: "fällt's". It's two seperate words, “fällt es", which can actually be combined BUT we need to use an apostrophe ' for it to work. It's actually used when we shorten a word and since the "e" is left out, we use '.

Also, nun ein paar spezifische Kategorien.

Well, I'm actually not sure. Just seems like the way to go with the comma. Also because there's a break between "Also" and "nun".

In Beziehungen verlass ich mich eher auf meine Gefühle, in aller W(sehr) wahrscheinlichkeit weil die Gefühle sich nicht einfach nicht bewältigen lassen.

"In aller Wahrscheinlichkeit" is not wrong, but it sound way too over the top. A summer form is appreciated.
I would switch "einfach" and "nicht" because it seems to make more sense to me. As in "they are not easy to handle" or if you actually meant "They can't be handled at all", then your sentence would be correct.

Ich glaube auch, dass das Gegenübersetzen von den Zweitellen der beiden teilweise falsch ist.

It was more of a stylistic issue for me. Your sentence was correct, but it sounds nicer this way. We use less words and "stellen" has a smoother tone than "setzen". It also refers more likely to something else rather than yourself, which is the case with "setzen". Not that it's wrong, but it could be replaced with something better.

Ich folge dien Wünschen meines Herzens soweit wie möglich, weil ich weiß, dass das Vernachlässigen des Herzens meinerseitsfür mich zur Unzufriedenheit führt.

"Den Wünschen" Plural and "Dativ". Not sure if there's a word for it in English, it's something that describes what type of article and adjustment we need to use. We use "Den" because we ask can "wem folgen wir?" And supposedly there's always an answer with dem, den, ihm, .... So mostly n or m at the end. Which leads us to the ending of "Wünsche" where we adjust it to the article and put an n behind.

Ab und zu erwünscht das Herz sich 'was Unrealistisches oder Falsches und in diesen Fällen muss ich bloß so hart wie möglich dasem Begehren widerstehen.

"erwünschen" is not necessarily wrong, but sounds over the top. So use a simpler version.
"Was" is in this case an abbreviation of "etwas" so we put an apostrophe because we shortened the word.
"Das Begehren" is alright, but it sounds better to say "dem Begehren" as we can also ask the question "wem oder was widerstehe ich?", Which is a Dativ question and Dativ refers to answer with, for example, "dem".

Mann muss immer wieder versuchen, Logik und Gefühle soweit wie es geht zu versöhngleichsam zu beachten.

A weird sentence, not really wrong, but not something you would say either. I think you meant something along the lines of "We have to consider both". In this case: "both logic and emotion need be considered alike".
"alike" is equal to "gleichsam".
"Zu beachten", "zu" because it is an "Infinitiv". A word that has not been changed from its original state and doesn't really refer to any subject or object, can be seen as the "to" of German. "Beachten" as in "to consider

Denn ohne das Achten auf Gefühle leidet mann unter an unerfüllten Träumen, und ohne das Achten auf Logik gerät man in Schwierigkeiten.

The aesthetically correct sentence would be "Denn ohne auf Gefühle und Logik zu achten, leidet man an unerfüllten Träumen und Problemen." It is shorter, and there where too many nouns. It's better to use a variety of words.
Otherwise, the only mistake was "Mann unter unerfüllte Träume". I assume "Mann" was a mistake because the second one is written with only one n, which is the correct use in this case.
"Unter" is alright, but normally we it's more likely used with singular nouns, so instead use "an" because there are multiple things (Träume ) that you suffer from.
"Unerfüllten Träumen" – it's not just plural, it doesn't have the usual article. It's in use with "an" so we gotta adjust the the words.