For an ICO, can we first get private investors for a 5% token ownership at a particular price, then hold a pre-ico at a slight higher rate and then an ICO for the public with an increased price?

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Of course I don't see why not. But u need to disclose this clearly in ur white paper for transparency and credibility purposes. Besides that nothing stops u from allocating your ico fundraising as u see it fit according to the above phases as long as u deliver on ur promises and offer a real value proposition to your potential investors
Hey Umar, if you're transparent with your funding and token distribution - why not? did something like you're suggesting. Statement from their team regarding funding and token distribution:
"• We received $1 million USD during a five-day public pre-sale in August.
• We have secured another $2.975 million USD in private investments.
• As of writing, we have raised around $300,000 USD from our current ICO, which runs until November 21, 2017."

Caveat: Some ICO participants don't like it when there was an initial private investment/ VC round as these investors might dump their discounted tokens once they are at an exchange. However, same could happen when a big discount is given the higher the contribution is, then you create "whales" that have an unfair advantage towards smaller investors that only contributed some ETH or so.
Possible solution: ICOs that have a KYC and an investment cap per day and discounts for early adoption but not for amount of contribution.
Absolutely! If you are the owner of the coin or tokeb, you can do whatever you want and structure it accordingly. Your idea to have a limited offering to qualified investors first and then pre ICO and then ICO is the way to go. You're on the right path.