I would like to know whether we can customize Sharepoint Online (Office 365) home page? Previously on MS Sharepoint 2010 & 2013, we can customize the masterpages and have our preferences Intranet portal, but I wonder this could be done in new version Sharepoint Online?
Yes, you can still apply your own master page in SharePoint Online! But it is no longer recommended! Microsoft is frequently rolling out updates that you'll miss out on when not using the built-in master page. Microsoft is on a journey towards a fully responsive design with the new modern look. You would bypass this on sites where you have applyied your own master page.
You can modify master pages on SharePoint Online by going to
Look and Feel -> Design manager
For more, read