Do I need to permanently keep my wallet open to act as a Holder?

Yes, your wallet has to be kept open 24 hours a day. This is really done in such a way that, if you secure your wallet, your Florin are 100% safe because your wallet will be encrypted all the time and only you can access it.
Of course a desktop uses a lot of power, a laptop less but still it is not very desirable to leave your laptop on all the time. Therefor there is also a Cloud version developed by the a community member.