Is it possible to download the new emails from the Gmail account while online, and then read them while offline? Is it possible to write emails while offline that will be sent when online?
Yes. And not just for the inbox, but inbox and sent may be the only folders synchronized by default.
From within the android gmail client: Press menu select more select settings scroll down and select labels change the number of days from which you want messages kept on the phone change individual labels (effectively folders) to none, this number of days, or all
NOTE: this answer was written about a now obsolete version of the gmail app, which continues to change in its user interface over time. The answer will not be updated to track that evolution, as there are multiple versions in use at any time which behave differently, and it is ultimately the idea rather than the specific steps which are important.
As far as I know, gmail will "cache" your most recent emails. If you have to "search" or browse through any labels, you probably won't be able to get to those emails. But your inbox (or at least the most recent in the inbox) should be available to view since your last sync. Additionally, you should also be able to draft emails while offline (and save the drafts), you just won't be able to send them until online again.
Is it possible to download the new emails from the Gmail account while online, and then read them while offline?
Yes, but only for your inbox I think. If you open your inbox on your phone while you have a connection and let all of the messages you have unread propagate to your phone, then close the Gmail app and put yourself in airplane mode you can reopen the Gmail app and read through your unread messages.
Is it possible to write emails while offline that will be sent when online?
Yes. If you compose and send an E-mail normally the messages that say "Sending..." will go out as soon as you have a connection again. I just tested this by using airplane mode.
The MailDroid app gives a lot of storage options to allow offline caching.