IOS – Do any iOS Email Clients allow the folders to expand/collapse like Outlook

You can configure Gmail as IMAP account type. That will sync your mail messages between multiple mail clients.

That leaves open your request to not retrieve all 3Gb of mail when first configuring the program. There I haven't got an answer only a possible workaround. Many mail clients limit their download sizes when they are on low bandwidth connections. You could try to limit your bandwidth, for exmample to 8 kbps, with these terminal commands when having Gmail configured to use secure imap:

$ sudo ipfw pipe 1 config bw 8kbit/s #create reduced bandwidth pipe
$ sudo ipfw add pipe 1 src-port imaps #sent traffic from imaps thru pipe

As Gmail is not offering an Exchange server, only ActiveSync compatible, you will never get an exact similar experience with Gmail.

Mac Mail saves the emails on the hard drive. I would try to find A mail app that doesnt save the mail to the hard drive but just sees the mail from the cloud email server.

If you use Gmail as a mail client for those emails then I recommend Mailplane. Others like Spark, CanaryMail, Mailspring and Polymail are all good alternatives along with others in the Mac App Store.

Other email clients should offer an option to store less emails offline and limit the size of your cache to a manageable size.

While there are steps to clean out the space for Mac Mail, I personally think signing out of it altogether and using an alternative might be best.