There is no Qlik hosted Cloud service, as there is for Sense, but you can spin up an Amazon AWS, Rackspace or similar server and install your QlikView Server on there. You need to ensure that you set the firewalls etc. correctly so your server can see out to your source data, but you keep your server secure.
A company called Rosslyn Analytics used to do hosted QlikView solutions, but I don't know if they still do this now that Sense is proving to be such a good option in the Cloud.
Yes, QlikView can be run from a hosted instance. There are many Qlik customers who are successfully running QlikView from the cloud.
There is an older document QlikView and the Cloud and AFAIK, it is still very much pertinent.
Hope this helps,