How to cash out E*Trade stocks (transfer funds from US to the EU) with Bitcoins?
*You basically need to get money into E*Trades US back account.*

E*Trade will either want a wire transfer (from Europe) or a bank transfer from the US.
They have to be very careful about Anti Money Laundering. So  wont want to accept money directly from a Bit-Coin Exchange.

You can open a US account from your local EU bank. Your local bank will act as a proxy to make the required AML and KYC checks. You can then use bitcoin to route the money via a bitcoin exchanges to that account. 

However if this is just a one off. You will probably find the costs, (fees at the EU exchange and the US one) and risk of bit-coin volatility; not worth the effort.  

Remember this is a problem which existed before bitcoin. So there will be ways of getting it to work with out bitcoin.

I would contact E*trade EU and US and ask them what would be the best way to find you purchase. (E*trade EU may have an arrangement to do this simply, already)

If you are buying a large enough quantity of shares, then the wire fees etc will only amount to a few percent anyway.

That being said having a US acc would be useful for BTC stuff later on. For example if want to work remotely for a US company, and need an efficient way of transferring money regularly to you home currency personal bank account.