As we can test, user interface elements set on the Cloud is limited to basic built in controllers with exception of Locators. If this is going to change is a subject of this discussion: Cloud Dynamic support. Do Buttons work in the Wolfram Cloud in Manipulate?

but since things are as they are, can I enchance my deployed cloud objects with javascript written by me? More precisely I want them both exchange information. One way communication is ok too, but by one way I mean from js to cloud object.

For example, Locators are not available, so can I add a script which captures cursor position and shares those values/variables with cloud object which is performing some procedures on them?

My first impression is, no, can't be done, but maybe it is possible, CDF plugin was able to read style properties of the tag it was embedded in.