In the title of your question you ask "What do you do when a moderator is saying insulting or belittling things?" and the answer to that is: the same thing you do if anybody else says insulting or belittling things, as addressed in the linked answer. You shouldn't avoid flagging something just because a moderator posted it; everybody errs sometimes, and when somebody does it should be flagged and addressed no matter who it is.
When somebody is being insulting it's best to flag and move on instead of engaging. Don't continue the argument in comments on the main site; that just makes things worse. Flag, or bring it to meta as you did here, or raise it some other way.
All users are expected to abide by the Be Nice policy. Moderators are users too, and we should be held equally accountable. Personally I believe that moderators, as caretakers of the community and holders of extra privileges, should be held to a higher behavior standard, though I believe that officially Stack Exchange does not do so.
In your question you ask what you should do to object to comment deletions. Comments in general are not supposed to stick around forever; they're meant to be temporary Post-It notes to request clarification or provide feedback that can be used to improve the post. They're not for discussion and arguments. Mi Yodeya has one of the highest comment-to-post ratios on the network (data from 2014). Lots of comments get deleted. So long as the deletions are fair -- for example, both sides of an argument are deleted -- this is not a problem. If you think comment-deletion has been overzealous, bring it to moderator attention for another look. You can use a custom flag on the post the comments were on.
If you feel that moderation is not being done fairly, then see this answer -- try flagging (several people see flags and we hope a moderator will avoid handling a flag when there is that much personal involvement), and if all else fails, you can contact Stack Exchange and somebody will take a look.