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Does the sequence of chord types (major, minor, diminished, augmented) in a key (major, minor, melodic minor etc) influence which chord progressions works well for that key type (e.g. a I-IV-V perhaps working better for major rather than minor), and if so, what about the chord type sequence of a given key type makes progressions weaker/stronger than another?
Musically you wouldn't say better or worse (this is usually based on opinion), but you may say Stronger vs Weaker.
A strong chord progression would be one that would generally follow the pattern: Tonic > Sub-Dominant > Dominant.
TONIC: I , III, VI
SUB DOMINANT: II, IV
DOMINANT: V, VII
I-IV-V is strong because the progression follows the order of (I) Tonic - (IV) Subdominant - (V) Dominant.
I-V-I-V is also strong even with just two chords because the chords are progressing: (I) Tonic - (V) Dominant - (I) Tonic - (V) Dominant.. There is no Sub-Dominant chord, but it is still progressing.
II-VII-I is strong because it is still following the progression: (II) Subdominant - (V) Dominant - (I) Tonic. Even though we didn't start with the Tonic chord, it still follows what would be considered a strong progression.
Given this simple algorithm, I can have a Computer just arbitrarily pick random chords that follow the order: T > SD > D without regard to key or voicing, and I should come up with what would be considered a strong chord progression.
I wrote a python script that randomly picked this: II - V - I - IV - VII - III - IV - V
This is an 8 bar progression that would psychologically sound fine to the normal listener.
Basically a weak progression would be one that doesn't follow this order, and you may hear the term Digression.
Examples: VII - II - III - IV or I - I - I - I or IV - II - I - III
When you master this basic type of composition, it is recommend to start incorporating more advanced harmony techniques like Secondary-Dominants, Modal Interchange, 6th chords, Chromatic Harmony, blah blah blah..
In a nut shell, yes there is an order that is more psychologically pleasing for the average human, and this is usually the order Tonic > Sub Dominant > Dominant. Without getting too technical, the reason is because the Dominant chords have 2 leading tones that naturally want to resolve to the Tonic chord - and our ears are pleased when the Dominant leads to the Tonic chord. When the Dominant chord goes to something else besides the Tonic; our ears notice this and say "WTF, that doesn't sound like the Tonic chord, there wasn't any resolution, this sounds like a weak chord progression, I may like it, but it sounds weak."
Again this is a technique used for composing and you may hear the term Circle of 5ths. Check out the Circle of 5ths if you haven't already done so and you may find a deeper insight to why this works.
I doubt that you can see that some sequence 'makes a progression better or worse than another'. This is something that is completely up to the composer and what he wants to achieve.
For instance, it is really common to use a major chord as V, no matter which scale you use. People like that because it has the leading tone (B) which resolves really smoothly to the tonic (C). But, there is no rule against using a minor chord as the V (like in the natural minor scale); if you like that sound, go for it. Here is an example that uses a minor chord as V:
Of course, each progression you choose has its own sound, so everyone will sound different. Also, it is common to borrow chords among the four aforementioned scales. So, you can start in the major scale with I and then borrow the iv from the natural minor scale and go back to I and so on; that was another really common trope used in songs.
