Can I use Gmail while offline? 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.