Is anyone allowed to share modified program using same name?
It depends; the GPL doesn't really specify that, since names aren't copyrighted. They are, however, obligated to clearly state it's not the same:
The work must carry prominent notices stating that you modified it, and giving a relevant date.
To control the name itself, you need a trademark, and you can certainly deny anyone the use of such trademark for forks of the program, even if it's GPL licensed. As an example, Firefox is distributed as Free Software (including GPL), yet you can't call modified versions "Firefox", due to Mozilla's trademark policy.
Do Ii have any ownership on software?
You, as the developer of the software? Yes, you are the copyright owner/holder; in fact, that's why you can license the software as GPL. As the copyright holder you are also not subject to the license yourself, and may distribute it in the future under a different license (although you cannot revoke the GPL licenses of versions previously distributed under it).
As an example, MySQL AB (one of the previous owners of MySQL) used to distribute MySQL as both GPL and sell commercial licenses to people or companies who didn't want to become subject to its restrictions.
But you should be careful: if you accept contributions from others and they don't assign their copyright to you, you are not the owner of such code; although you can still use it under the terms of the GPL, you can not re-license it.
Do i take any responsibility for program usage, unexpected behavior,...?
Not because of the GPL; it even includes sections disclaiming warranty and liability. However, a copyright license doesn't supersede the law, so you may still be liable. You should check with a lawyer to know the relevant legislation on that matter.