Is it good to be courteous? Yes. And you should be courteous to people of both sexes. This includes holding doors open for people, carrying items for people when appropriate and given permission, and so forth.
Now, I have experienced a few different situations that involve opening doors, carrying things, etc. where a guy is being nice, and a guy is being a jerk. It all depends on the context of what actually happened.
Scenario 1: I am walking up the hallway, carrying a box that is obviously heavy for me--i.e., I am struggling to hold onto it, stopping to take breaks, etc. A man walks up to me and says, "Excuse me, it looks like you're having trouble with that. Can I take that for you?" I smile with gratitude at the thoughtful person and hand them the box (or carry it with their help).
This is a lovely thing to do, and of course the man is not being sexist. The two important factors here is that a) I am legitimately and obviously in need of aid, and b) he asked to be sure I was okay with his involvement, and c) he's made no (shared) judgments about me regarding my gender or ability to carry a box, etc.
Scenario 2: I am walking up the hallway, carrying a box that I appear to be managing quite fine on my own (keeping a brisk pace, no struggling, etc.). A man stops me in the hall and says, "You're a woman, that's too heavy for you, let me take that," and he proceeds to try to take the box out of my hands without my permission. I proceed to hug the box to myself (he's trying to take something from me without asking), and say, "Excuse me, I've got it," and push past him and move away as quickly as possible. The man mutters, "I was just being a gentleman! *****." ((NOTE: this scenario pretty much has happened to me IRL))
The important factors here are that a) Although I showed no signs of needing help, he presumed I was incapable of carrying something because of my sex, b) he started to take my belongings away from me, and c) He approached me and violated my personal space without permission or invitation. All of these things are the antithesis of being gentlemanly, courteous, chivalrous, or civil in any way.
And in my personal experience, usually I find someone who calls himself a "gentleman" (or gentlewoman/lady for that matter) isn't. Because they're drawing attention to their behavior and expecting to be rewarded for it---showing there is an ulterior motive which has nothing to do with being respectful of other people.
In short: show respect for people and their dignity. Observe the individual rather than make assumptions based on broad categories, ask rather than presume, and never invade someone's personal space. As long as you do that, you are a "gentleman" or "lady."