Moving email from Hotmail to Gmail

I am a full on apple user; MacBook Pro, iPad, iPhone, iCloud etc and I just made the move from Hotmail to Gmail today so that I can use IMAP since I had problems with syncing of mail between Mac Mail app and my iPhone, iPad etc.

My question is now that Gmail has default folders (important, starred etc) and all of those emails don't show in my "inbox" on my iPhone but they all show up in Mac Mail. How do I get all the emails to appear in my iPhone inbox?

Secondly, my intent is to still retain the use of my hotmail email address since all of my email contacts know that address. What I did was; import my hotmail to my gmail through the gmail web application and then set up the "check mail from other accounts" to pull email from hotmail into gmail so that I can basically run my hotmail account through gmail. I wanted IMAP functionality so that there is syncronization of mail whether i'm using mail on my macbook, mail on my iPad or mail on my iPhone.

Is this the correct way to set this up so that I am still using hotmail but running it via gmail? When I reply to an email on my phone through this new gmail account, it replies from the gmail account instead of the hotmail account...how do I change that?

Thanks for your help.