"I have a cat because I love cats."
Translation:יש לי חתול כי אני אוהב חתולים.
That's interesting, someone in another discussion said the exact same thing yesterday, and I don't think I've ever been told that בגלל has negative connotations! I've been using it as neutral by whole life.
As for בגלל ש, it's formally incorrect and I wouldn't use it on an exam. But it's in very common daily usage, enough to make it a part of spoken language. It's sort of incorrect in the same way that, in English, a preposition at the end of a sentence is something up with which we should not put. You know, you are told in class never to use it, and then you go out and use it anyway.
The "no ending with preposition" is a myth though. Actually concoctions like "up with which we should not put" seem to be used by grammarians to demonstrate the absurdity of this "rule".
A better way to compare formal English to colloquial English would be "who" vs "whom" as objective pronoun.
WRT negative connotations, yes, I was definitely taught בזכות as the term to use if there were positive connotations, and בגלל if there are negative. If the sentence here were neutral I could understand using בגלל, but since loving cats is presumably positive, בגלל would have been marked wrong by my ulpan teacher.
WRT בגלל ש, does DuoLingo give the option of having the sentence with בגלל ש accepted if someone types it in, but not including it on the list of sentences for the exercises in which you need to select all the correct translations? That seems like a reasonable response to the "it's not actually correct, but it's common usage."
As far as I know, בגלל is neutral - it can be used either for negative or positive sentences (you can find examples in the Bible for positive uses). There are other negative synonyms - מחמת, מפאת, בעטיו של and others.
Regarding בגלל ש, it's not formally incorrect, see for example here - https://www.safa-ivrit.org/style/biglal.php . I'd also guess it's the second most used word for "because" after "כי" in spoken language.
ll כי can never be fused to words. There is, however, the preposition -כ, which can replace the word כמו (like, as), or in other contexts mean 'around, about'. For example:
Clear as day = ברור כשמש (though literally, "clear as the sun")
I have about a hundred stamps = יש לי כמאה בולים
It's more common to use כמו rather than -כ in speech, as -כ is considered more formal.
As for כאני, that wouldn't work anyway: for "like me", you'd only use כמו with the pronominal suffix, i.e. כמוני (kamoni).
According to Muraoka (Modern Hebrew for Biblical Scholars, xlii), the classical Hebrew כי is still in use (obviously as the sentence here indicates) as a causal conjunction "though always following the main clause." He doesn't write anything about בגלל having negative connotations. I don't think the latter (whether as conjunction or preposition) has negative connotations because none of the modern Hebrew grammars I looked at say anything about that. They do tend to say that בגלל can occur at the beginning or middle of a sentence and is followed by a noun (G. Etzion, Routledge Intro Hebrew Course, 323). He gives as examples: בגלל הרעש, "because of the noise," and בגלל הילדים, "because of the children." If there are any negative connotations related to a specific בגלל + noun construction, it is coincidental rather than grammatical. Etzion also notes that בגלל takes suffixes and so הכול בגללך, "it's all your fault," "it's all because of you," and בגללי, "because of me," i.e., I'm to blame. Those are obviously negative, and so probably Hebrew teachers will sometimes help students with this latter construction by noting the negativity, which is useful, but in wider context the preposition and conjunction are not negative.