Can you say "feel to" do something?
I have come across this usage of the catenation feel + to-infinitive at Church ('I felt to ask him...'; 'I felt to go to ...') but consider it to be non-standard. I can't find dictionary or Google Ngram support for the catenative usage. Perhaps it is a downtoning of 'I felt led to ask him...' here.
The feel + that-clause
I don't feel [that] I can trust him.
I felt [ ] I ought to ask him.
(whether the 'that' is included or not) is quite acceptable, of course.
I'd say, "I don't feel like trusting him."
I think I would say 'I do not feel inclined to trust him'. I can understand why the Italian might be different. Most would agree that the Italians are more 'touchy-feely' than the English. What a fabulous place it is. Lucky you!