Let's say you've signed up for an Office 365 developer account. The following is what you get access to:
Q: Can I download Word, Excel, PowerPoint, Outlook, OneNote for free now?
A: Yes, click on the red "Install now". I'm not sure if your developer account allows you to use them for the duration of your developer trial, but I'm pretty sure they won't work after your 30-day trial expires anyway. This isn't a way to get Microsoft Office for free. The point of having those apps there for you to download is to show that for any Office 365 user, they get installations of the Office apps that comes with their Office 365 subscription that they either purchased themselves or that their company purchased for them.
Q: Why does my "OneDrive" or "Sites" icon remain grayed out even after a few minutes?
A: SharePoint is still provisioning. When you set up an Office 365 tenant (remember: a tenant looks like @abc.onmicrosoft.com or @abc.com), you are saying that I'm buying Office 365 for all the users belonging to my company which has a domain name of abc.onmicrosoft.com or abc.com. Office 365 doesn't only include Word, Excel, PowerPoint etc., it also includes SharePoint (a powerful intranet/file management and collaboration solution) and Exchange (an email server that allows you to provide email services for your company). SharePoint and Exchange used to be sold as software in boxes, but now they are all online as part of the Office 365 cloud solution. The icons "Sites" and "OneDrive" represents SharePoint. In particular, it may take some time for SharePoint for your abc.onmicrosoft.com or abc.com company to set up so be patient.
Q: It still remains grayed out even after 1 hour!
A: SharePoint probably failed to provision. This is something that we are aware of and working on fixing. Please post your problem on the community forums (click on "?" in the top right hand corner) and someone will work with you to fix your specific problem.
Q: What is the difference between OneDrive here and OneDrive.com?
A: "OneDrive for Business" is distinct from "OneDrive.com" and "OneDrive". The Pro/Business solution is essentially a file management solution from SharePoint that is a cloud storage solution for your company. OneDrive.com is cloud storage for individuals. Let's say you are working on an important proposal for a meeting next week at work. That would probably go onto OneDrive Pro and not OneDrive.com. But if you want to store your own photos taken at your kids' birthday party, that would go onto OneDrive.com. In the context of Office 365 tenants and developer accounts, the word "OneDrive" refers to "OneDrive for Business". Outside the context of the enterprise, the word "OneDrive" refers to "OneDrive.com" or OneDrive for consumer services.
It's confusing, I know. But I'm here to answer your questions - feel free to post them in the comments below.