I'm using a "QWERTY" Hebrew keyboard, so my W key is shin/sin. I just looked again and the key directly above the apostrophe also looks just like an apostrophe, but upon typing the two next to each other they are actually different. I'll try it next time and see what happens.
I am not a native israeli, but I have been learning hebrew for quite a while, and I could imagine saying לי יש זקט if you're specifically emphasizing the "לי". Like if people are saying, "I have a coat", "I have a scarf" etc., I'd think you could then say, "לי יש זקט" -- me, I have a jacket" (but grammatically sounding better than my sentence does in english...).
Fair question. Indeed, this would take a rather specific set of circumstances. It might almost call for punctuation that highlights and clarifies the situation.
(Something like - ".לי - יש ז'קט")
This might be the sort of context where, indeed, the matter of jackets ('with the indefinite article' as it were) has already been brought up in the discussion, and it is specifically the matter of ownership that's being addressed. (along the lines of "... as for me, why, yes indeed, I do have a jacket.")
A context could be imagined, similarly:
"I have a jacket ... but she doesn't [have one]."
That would most definitely lend itself to being translated as:
".[לי - יש ז'קט, אבל לה - אין [אחד"
If both words came into Hebrew from English, your question would be valid. However, ז'קט is a word that came into Hebrew from the French word jaquet, which is the same as jacket. However, French and English pronounce the "j" differently. That's why they are different letters in Hebrew. Remember, not every Hebrew word that resembles an English word came from English.
What you have there literally translates as "the children there is a jacket". The lamed prefix makes it say "TO the children there is a jacket". Putting the יש at the front of the sentence (and still using the lamed prefix) would translate as "there is to the children a jacket".