Translation:Are there yellow flowers blooming in the yard?
A very good sentence to explain word forms where the four い functions differently and these functions are common.
The sentence is read にわに (きいろい 花)が （さいて いますか) [or ... (さいて います) か].
The い in きいろ is a part of the noun いろ [colour]. Native nouns usually have readings of several kana (shorter ones like め, て [as hand] are exceptions). In this case, a single い does not make sense.
The second い is the adjective ending. Technically it is 連体形, modifying the flower.
The third い is from the special form of the 5-dan verb さく (to bloom). In classical Japanese it was さきて, and the /k/ in き was dropped and the form becomes the modern さいて.
The last い is from the auxiliary verb いる, which means 'is doing'.
On いろ. There is an (arranging) order for kana as iroha-order, which was mainly used before mid 20th century and is still being used today (though far less prevalently as the gojyuon-order). It stemmed from an ancient poem いろは歌 (いろはにほへと..., though colourful flowers bloom in a charming scent...), where all the 47 classical kana were used, and each kana used exactly once.
On 5-dan verbs. The rule is /k/ /g/ dropped (so い remained), /r/ /h/(as in 買います [which was 買ひます in the classical orthography]) /t/ to a little っ, and in the case of /n/ /b/ /m/, /i/ is dropped and the consonant is reduced to ん. In the case of /g/ and /n/ /b/ /m/, the particle て changes to で, た to だ.
I was expecting the answer to be 'are the yellow flowers in the yard blooming?' Would that be the answer if the article 'が' was a 'は' instead?
No. That would be: Niwa ni aru kiiroi hana ga saite imasu ka. I have no Japanese keyboard on this device.
It turns にわにある ("in the yard") into a descriptive phrase modifying the noun phrase that follows it, きいろいはな ("yellow flower(s)").
but what exactly is ある? a verb, a conjunction, a preposition? I tried but couldn't find its definition and function anywhere.
ある is a verb meaning "to exist"/"to be" for inanimate objects. (いる is its counterpart for animate/living things.)
No, the "ga" is not the problem here. We use another form to say something like that.. I think it's the possesive form. It will be: niwa no kiroi hana wa saite imasuka? = are the yellow flowers in the garden blooming?
I don't know about the yard, but I know there are yellow flowers blooming inside an abandoned church.