Beer has been made for at least 4000 years, but yeast was discovered in a microscope in the 1600s.

Yeast was not understood to be a microscopic organism until the 19th century.

How was beer made before people knew they had to put yeast into the wort...Do we have any contemporary sources on what they thought caused fermentation?

People did not know yeast was a microscopic organism but they knew yeast caused fermentation. Brewers would save the "foam" from one batch to make the next. In a similar fashion bakers would have a some sourdough starter from one batch to make the next. In both cases they are maintaining colonies of a particular yeast named Saccharomyces cerevisiae. Saccharomyces cerevisiae (or its close cousins) is the species of yeast that excels in turning sugar into ethyl alcohol.

I know wild yeast is everywhere, but that can often produce off flavors in beer. Was there a lot of inconsistency in early brewing?

Judging by all the superstitions involved in brewing its safe to assume the fermentation or aging failure was not uncommon.

Most flat out spoilage actually came from bacteria as opposed to errant yeast however. The main "spoilage" yeast is brettanomyces but in many areas the effects of brettanomyces were considered a welcome addition to a beer's character.

I've also heard about brewing sticks that would preserve a yeast strain and would be passed down for generations. Is there any truth to that?

Ive heard that about Scandinavian brewers (if I am remembering correctly) but yeast gets in everything (including the brewers themselves) so the sticks wouldn't necessarily be needed. Its a moot point though since specific colonies of yeast were maintained by saving the "foam".