Shall I use my university address to contact professors in other universities who are specialized in the field of my dissertation?
I personally use my university address any time that I am conducting academic business, as, in my experience, people place more trust in .edu email addresses and web domains. Also, it is more professional to use a university email address as opposed to a personal email address.
However, keep in mind that your university likely has access to your email account, and they could track what you are saying and to whom. So, if you need confidentiality from your university, it may be better to go with another email provider. Also, some universities will 'expire' email accounts when students graduate (or faculty/staff leave). This likely won't be an issue for you, though, as you are still working on your dissertation.