Out of one side of our audiophile mouths we talk about how bad home theater receivers and processors are because of all the things jammed into them, but then we rave about how great it is to have an amplifier, preamp, headphone amp, and phono stage in an integrated. Hmm.
There's no one size fits all "right" or "wrong" answer to this question.
Separates typically give you more flexibility. They usually have more input/output options. If you decide there's a weak link in your system (you need more power, you need a phono stage with more gain or loading options than the one in your preamp, the headphone amp doesn't have enough power for your new cans, etc.), you may be faced with replacing your integrated or adding another component that duplicates one of the components in your integrated.
On the other hand, an integrated may have all the features / functionality you need and require less space. You don't have to get anything more than the amp / preamp in the integrated and can add other items as needed.
I have 2 systems with separates and two with integrated amps. I use what I want/need that works best in the space I have and gives me the features/functionality I want/need in that space.
Whichever option works best in your space and meets your needs is the "best".
Honestly, it sounds like you're in the early part of the learning curve. You would probably benefit from finding a couple of dealers near you that have good reviews and spending some time describing your listening environment, what you currently have, and what you want and getting some recommendations. Your questions are pretty broad and you would benefit more from an extended conversation that a few posts on a message board.
