"Raj is very happy."
Translation:राज बहुत ख़ुश है।
It represents the 'ch' sound in the Scottish 'loch' and is mostly found in words borrowed from Arabic/Persian. If I am not mistaken, it is represented by its own letter in the Urdu Nastaliq script. However, many Hindi speakers drop the dot (called 'nuqta') and pronounce/write it as ख (this is true for most dotted consonants with the exception of ड़ and ढ़).
ख is an aspirated form of क, and ख़ is similar but from further back, the first two are gutteral, but with ़ is from the epiglottis. (See Wikipedia 'place of articulation' and 'devanagari#consonants'.)
It's usually transliterated as kha with underline, but yes per above it's essentially aspirated क़ - I'm not sure why that's not transliterated as ka with underline, or ख़ as qha (as you wrote), but there we go!
ख़ is not the aspirated form of क़. They both represent different letters in Arabian/Persian.
What you said about the place of articulation being further back is true for क़.
Since this letter is found only in Persian/Arabic loanwords and these languages do not distinguish aspirated and unaspirated consonants, the aspirated form of क़ is absent in Hindi.
On the other hand, ख़ has the same place of articulation as क and ख. It's just that for क and ख, the tongue actually touches the soft palate (they are plosives) while for ख़, the tongue almost touches and you force air through the narrow opening between the tongue and the soft palate (it's a fricative).
Khaa with a dot is the hindi equivalent of the persian/arabic gutteral kha, as in loch. This urdu/persian/arabic kha looks like the arabic emphatic "haai" but with a dot at the top. If you go to the arabic letter section you will see and here what I am taking about.
Anyway, as this sound does not exist in hindi and is difficult for many Indians to pronounce, it is noted as a different sound/letter with the dot, and then just ignored when spoken to be pronounced as a kaa.
Hindi speakers with greater familialarity with Arabic/Persian loanwords, or "Urdu" words, ( and who have taken the difficult task of learning how to pronounce this gutteral "kha") will take note and pronounce it accordingly.
The non aspirated "ka" with a dot is the Arabic/Persian/Urdu qaaf, as in Qiraat, Quran, etc. It looks like the Arabic "faa" but with two dots at the top in stead of one.
The aspirated "Kaa" with the dot, found in "kush hai" is equivalent to the gutteral "khaai" of Arabic/Persian/Urdu, as in Khalid,Khaleej, or Khushboo. Again it looks like the Arabic emphatic " haai" but with a dot at the top.