Yishtabach does not start with a beracha because it is adjacent to baruch sheamar, and both were instituted for pesukei dezimra - one before, and one after.
One does not say amen to "melekh mehulal batishbahot" at the end of baruch sheamar, but instead after "hai haolamim" at the end of yishtabach, which is the end of the berakhah.
One who speaks between Yishtabah and Yotzer - it is a sin in his hand, and would be required to return from the battlefield [due to fear of sin leading to death] (see Deut. 24, Sotah 44b). But there are those who say that it is permitted to interrupt at that point for communal needs or to bestow charity to those who come to be provided for from charity. Gloss: From this stems the custom in many places to say a blessing for the sick or for a claimant to ask for judgement between Yishtabah and Yotzer, because these are instances of (interruptions for) the purpose of a mitzvah. And after that, when they resume praying, the leader says a bit of pesukei dezimra and then kaddish on those verses, since we do not say kaddish without a psalm before it. This is why the evening service begins without a kaddish (Kol Bo), and why one who did not have a tallit or tefillin at the start of the (morning) service and one was brought to them between Yishtabah and kaddish, can put them on and say the blessings. But between kaddish and Barkhu, one must not stop for any purpose (Or Zarua), and all the more so not stop after the leader has said Barkhu and before they begin saying the Yotzer blessing (Beit Yosef).