If I have a status on one member airline in Star Alliance (SA), do I automatically get the same status on other SA airlines?
If I don't get it automatically, can it be requested?
No, you don't technically get the same status on other airlines. And it can't be requested.
How it works is that Star Alliance has 2 generic levels of status - Star Alliance Gold and Star Alliance Silver. Each particular airline status corresponds to one of these two statuses.
For example, if you're a Silver at Continental, you're also a Star Alliance Silver member. If you're Platinum or Gold at Continental, you're a Star Alliance Gold member. You can check out the equivalent for your particular airline at this page: http://www.staralliance.com/en/benefits/frequent-flyer/
What this means is that, if you have your frequent flyer number entered on all your reservations, when you fly with other airlines in Star Alliance, they will see that you are Gold/Silver and give you benefits accordingly (lounge access, priority boarding, fast track). You can also earn miles with your own airline whenever you fly a Star Alliance member airline.
However, this isn't the same as being a member of the other airline's FFP. For one thing, airlines often restrict reward availability to partner rewards and retain some of the inventory for their own members.
E: I wanted to add that you can technically ask for status on other airlines, if you are planning to switch all your flying to them. You can contact them and ask for a "status match", which they will give you if they think you're a high value passenger, and usually you'll have to fly a certain amount within a certain number of weeks to retain the status.