I have sent out multiple Dropbox invitations to people who have subsequently created a web account from my invitation but have not installed it on their computer. In the Dropbox referrals list, each person's description says "waiting to install". I know I won't get the extra GB unless they install it but does that also mean they can't share a folder with me?
They can still share folders and files with you through the web interface, and vice versa - you can share folders and files with them, which they can view through the aforementioned webapp. Only difference is they won't have those files stored locally (unless they choose to manually download them).
Yes, it does. Dropbox's system is secure and proprietary; they need to install the software before they can share files with you.