How do I use mSIGNA without the standard Bitcoin client? Which IP do I have to fill in in mSIGNA to connect to the Bitcoin Network without using the BitcoinQt/bitcoind application?

Is it possible to use mSIGNA and BitcoinQt even if the syncing process is not finished yet?
Yes there is a way, you need a way to connect to someone running BitcoinQT and who has made their node public. BitcoinQT runs as a node, but to make it discoverable to any but other BitcoinQT wallet users you have to go a step further. 

So you need a list of such BitcoinQT "servers", and I found one earlier and in searching for that list of open Bitcoin-QT "servers" someone had published I found this. 

If I manage to find it again, I am trying to solve my problem with sending from mSIGNA, I will post a link and more complete answer here.