Either way is OK
Either method is OK and both work. The best approach is to probably do some of each. Does your company have a preferred standard method?
Creating the Parts first and then assembling them into a Product is called "bottom-up" design. A disadvantage is when you work on an individual part you don't have the other parts available for reference.
Creating the Parts within a Product is called "top-down" or "contextual design". An advantage is you can design parts based on other parts. Another advantage is creating associative links between mating parts (although this could cause problems if you don't know (or pay attention to) what you're doing). This also requires that the parts designed in the context of the Product assembly should also be modified in the context of the same assembly.
If you look closely at the Product tree showing all the instances of Parts, you will see that the icons in front of the Parts have different symbols with different colored gears (yellow, green, brown, white) which indicates that you are "designing in context."
Both methods require the use of assembly Constraints to position the parts with respect to each other. I'm not sure why the constraints are causing your parts to "collapse" - unless you are editing or renaming them?
Using Publications would probably help avoid some of the constraint problems.
