I want to build a house in Falkreath. Does that mean I have to help the thane/jarl and the townspeople before I can build it?

You'll have to help the jarl with a quest to clear out some bandits. If you speak to him before he sends you a letter summoning you, you'll have to bring him some Black-Briar mead before he'll give you the quest to clear out the bandits. After you finish that quest, you'll get permission to buy land from the steward. In order to become thane and get a housecarl, you'll have to additionally help 3 people in the hold. This part is easy: sell the woman at Half-Moon Mill some firewood, pick crops and sell them to a farmer, and I think Thadgeir has a minor fetch quest that won't even take you out of the town proper.