Example 1
ふうせんがほしい (I want a balloon)
Simple enough, i assume that it is putting emphasis on the balloon being the want. I would assume (and am probably wrong in doing so) that you could say the same thing as, "ふうせんはほしい。" And I'm sure I'm wrong...
ほしい means want but it works (conjugate) like i-adjective.
When using ほしい as complement, it normally requires subject, and subject marker is が.
It is possible to say ふうせんはほしい but in this case it implies that there are some other options besides a balloon, the speaker want a balloon but may not want, say, origami.
(While being a topic marker, は has connotation that there are more than one options and the speaker is picking just one of them.)
e.g.
「ふうせんと折り紙のセットをあげましょう」
「ふうせんはほしいけど、折り紙はほしくない。 」
Example 2
わたしのほしいいろがない (They don't have the color i want).
わたしのほしい色がない can be translated into English as
they don't have the colour I want, but ない actually means "does not exist".
If literally translated, "there isn't the colour I want".
Same as ある.
I think it is a particular way of expression in English to use "have" (meaning possession) instead of using word meaning existence.
This makes me think of ほしい as more of a noun that's conjugated than an adjective. Making me think of "わたしのほしい" as "my want." Even then, i would want to put a じゃ between ほしい and ない and move いろが to the front.
No, it is not a noun, and it does conjugate.
To modify a noun, verb/adjective are used in its basic form.
But it does not mean verb/adjective do not conjugate.
e.g.
私はこの本がほしい → 私がほしい本
この車はゆっくり走る → ゆっくり走る車
あのネコは眠っています → 眠っているネコ
I don't really understand what you mean by the last sentence.
私のいろがほしいじゃない?
It does not make sense at all.
Example 3
あかいふうせんのほしいこどもがおおい (There are many kids who want a red balloon)
Personally, i would interpret this as "The want of the red balloon is many kids."
Are these just bad examples, or am i right in my assumptions?
あかいふうせんのほしいこども means children who want a red balloon.l
If literally translated, it should be "children who want a red balloon are many."
ほしい is not a noun. It is children, not "want", which are many.
あります example
水がありますか。
It's supposed to (i think) be interpreted as "Do you have some water?" Yet, が implies that 水 is the subject. Therefor, i would interpret this as "The water has...?" Is ある and いる mean "To be had by" or is this an example of an exception to が?
As mentioned under "example 2", ある and ない means to exist, not to exist.
Trying to analyse Japanese with English grammar/structure can be misleading.
(I think this is actually
your theory, if I remember correctly what you've posted some time ago.)
You should forget about "to have" when thinking about ある and ない.