Will there be any tools to cross-check ethereum records and the corresponding BTC transactions? Maybe some transaction ID's that will be the same in both blockchains?
I see no problems in embedding this into blockchain explorer. A script that will go through new blocks of bitcoin every X minutes and update cache with the most recent transactions mapping. We already did similar things several times.
Did you consider another blockchains except Ethereum?
One of the core ideas of the project is to use smart contracts instead of legal contracts. Therefore we consider any blockchain that supports smart contracts. So far Ethereum is the most mature project with this feature. If something better shows up - we'll consider using it instead.
What Ethereum benefits exist if its smart contracts don't control the funds but monitor them only?
This is not exactly correct. Smart contracts control the funds, but for now only in Ethereum network. We need Ethereum to implement insurance policies logic, risk models, interaction with oracles etc. Financial transactions is just a tiny part of the project.