The speaker claims chivalry isn't dead, only in remission, and then compares it to a cockroach sleeping under a stove in Hell's Kitchen. It's an unflattering image: chivalry survives, but as something grubby and hidden. From there the poem lists the gentlemanly gestures the speaker is willing to make: holding a door, carrying books, sending wine. Each one comes with a catch. The woman speeds past without noticing, nobody reads books anymore, and she hates wine. The gestures are polite but pointless, and the speaker seems to know it.
The tone is wry and a little self-mocking. Even the offer of protection is "this week only," and she towers over him in her heels. The closing image of chivalry as a coat we shed "to avoid the heat" suggests it gets dropped whenever it becomes inconvenient. The mood is amused but quietly rueful, as if the speaker is both defending old manners and admitting how empty they can look.