For the purpose of OD, SS/VE is just melee w/ different elemental capacity than VE/VE. I can't think of many (any) times when you'd use Sharpshot head over Valoredge or Soulsoother on Valoredge frame though as it potentially makes the puppet much more annoying to use in addition to giving you little benefit in elemental loadout. The more common swap is the Sharpshot frame over Valoredge in order to change the puppet over to piercing damage. You're going to use the VE OD set for melee (slashing/blunt), SS for ranged (piercing).
So since this info is actually surprisingly difficult to find in an easy to digest format, a brief description of what the parts do:
Range is primarily determined by your animator these days since your main animator will be the P +1 line. Animator P forces the puppet into melee range. Animator P II causes the puppet to maintain distance from the target that it's deployed against.
Heads are mostly swapped for elemental capacity however they affect spellcasting behavior for mage frames.
Sharpshot head causes the puppet to focus on maintaining enfeebles.
Soulsoother reduces cure interval and changes the % hp for the target that the puppet will cast cure at.
Spiritreaver focuses on offensive magic.
Stormwaker is more or less an amalgamation of Soulsoother, Spiritreaver, and Sharpshot: red mage basically. Will try to use every category of magic however it won't do any of them as well as a dedicated head besides Enhancing, which it excels at. It is the only head that can cast Phalanx and Stoneskin, and it will cast them on the master.
Valoredge essentially works like a worse Soulsoother. Worth noting that it can also cause the puppet to want to engage in melee combat on frames that normally don't try to chase the mob.
Generally the interval for each type of magic is changed based on the category of the head. Honestly you won't be using this feature very often and the one you'd be most interested in using 90% of the time is Soulsoother, 8% Stormwaker, 2% Spiritreaver for a mage automaton.
Bodies affect damage type, available weaponskills, spell pool, natural DT-%, and basic behavior.
Valoredge has the highest natural overall DT and deals slashing damage, and it will always try to stay in melee range and has no spellcasting ability.
Sharpshot has the same natural DT as VE, deals piercing damage, and will stay put when deployed unless influenced by an animator. It also has no spellcasting ability.
Stormwaker has 0 natural -PDT but by far the highest natural -MDT/BDT, stays put when deployed like the Sharpshot frame, and has access to a wide variety of spells whose usage changed depending on which head you attach.
Harlequin has about half the natural DT of SS/VE, the same weaponskills as Stormwaker, will try to stay in melee range, and can cast a limited selection of spells. It also has much less MP than the Stormwaker frame.
If you want specifics including how each head prioritizes spellcasting, how each head can change range behaviors on certain bodies (largely irrelevant these days since the Animator mostly overrides this), etc then it's all listed on the BG Wiki page under frames.