Did the Declaration of Independence inspire other nations to seek Independence?

Yes. The most obvious one is France, which underwent a revolution of its own against its own king and queen over a ten year period, from 1789 to 1799. During that period, France went from an absolute monarchy to a republic. In 1789 the French National Constituent Assembly used the Declaration of Independence as a guide to drafting its own "Declaration of Rights of Man and of the Citizen." Thomas Jefferson even helped write it while he was in France. The Assembly first wanted to make clear the rights that citizens had prior to writing a formal constitution. As a sidelight, one point made in the Declaration of Rights of Man was a condemnation of the notion of the "divine right of kings." This was a philosophy that Thomas Paine had written about in "Common Sense", which greatly influenced the French people.