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I know that every key has a set of 7 chords by taking each note of the scale as a root. I also know that the pattern for naming these chords go as follows: Major, minor, minor,major,major,minor and diminished.

Apart from these chords, can someone please explain how one arrives at the seventh chords? When googling, for example, chords in key of D major, I see the 1 chord as Dmaj, Dmaj7... and so on. Can someone explain how these 7th chords fit? Also, why is the v chord not an Amaj7 but an A7 (unlike Dmaj7, Em7, F#m7)? And also why/how is the vii° is C#m7b5?

You're referencing what we call the diatonic seventh chords. What we mean by "diatonic" is that these are the seventh chords that are created by using only the pitches in the prevailing key (and thus no chromatic tones).

As such, to determine any diatonic seventh chord, you simply take the root and find the third, fifth, and seventh above that note. But when you do so, make sure you use the pitches that are in the given key (in this case, D major).

So if D is your root, the remaining tones will be F♯ (the third), A (the fifth), and C♯ (the seventh). The F and C both have sharps because those are the versions of F and C that are in D major. When we take this four pitches, we see that this creates a major-seventh chord, and thus the chord is Dmaj7.

Contrast this with the seventh chord built on A. Here you have C♯ (the third), E (the fifth), and G. You were expecting it to be Amaj7, but that would require a G♯. But since there's no G♯ in the key of D major, this chord must use G♮ instead, and thus it's an A7 chord and not an Amaj7.

And the same is true for the C♯ chord: C♯ E G B results in the C♯m7♭5, also known as C♯ø7.

This is probably a dupe question, but it's simply 1,3,5, then 7, diatonically. In D, specifically, the V chord is A7 - A C# E G, and the seventh out of the list is C#m7b5 because it contains C#, E, G and B. The C# E and B giving C#m7, but because of the G, not G#, which isn't in key D, it becomes flat 5.

"why is the v chord not an Amaj7 but an A7"

Because it has a natural G, not a G#. That makes the seventh of the chord a minor seventh, not a major seventh, so, it can't be a major seventh chord.

Yet it has a C#, not a natural C. That makes the third of the chord a major third, not a minor third, so, it can't be a minor 7th chord either.

It's neither maj7 nor m7. It's the dominant 7th chord in the key of D. Dominant has arguably the single most important harmonic functionality in a key, and it's neither major nor minor.

Any 7th chord without additional identification as major or minor or something else (aug, dim, or weirder stuff) is nearly universally understood to mean a dominant 7th chord. It's normally created from the 5th scale degree in a major key.

That's not the only place dominant 7th chords are found, but other dominant 7th chords are either constructed chromatically (not diatonic to a key), or else modal constructs - for example, the tonic 7th chord in Mixolydian mode is this type of 7th chord (major 3rd, minor 7th). It's constructed the same way as a dominant 7th chord but in that mode it doesn't have the harmonic function of the dominant. This is beyond the scope of your question though.

Short answer: A7 is neither a major-7 nor minor-7 chord. It's the chord built from the 5th scale degree of the key of D major, adding the diatonic notes in that key a 3rd, a 5th and a 7th above that root (A).