Do bass pickups matter?

Yes. Bass Pickups definitely matter in determining your bass sound. Pickups are one of the most important areas to consider when it comes to shaping your tone, and swapping out your default pickups with a new set can drastically affect your sound.

How many pickups should a bass guitar have?

The amount of pickups a bass guitar has can vary depending on the model. Some bass guitars come with only a single pickup, others can come with 2 or 3.

Can I buy a humbucking bass pickup that fits in the space for a single coil? I want a thicker sound.

If you want a thicker sound, you may want to check out something like the Seymour Duncan SJB-3B Quarter Pound Jazz Bass pickup. Although still a single coil picker, this model is very high output and delivers a thick, hot sound that is very similar to what you'd expect to hear from a humbucker.

Are there benefits to changing my stock passive bass pickups for an active setup?

Yes, you will certainly change the sound! Active pickups tend to be both powerful and controllable, both of which are very useful. Also, they tend to sound modern so if you have a contemporary sound, active pickups may be a great idea! Active electronics in the tone control department will also open new doors for your tone. Vintage tones may not be readily available but that fact may help state a case for another bass guitar purchase...

Are bass pickups different from guitar pickups?

Yes. Bass pickups are sized differently to guitar pickups. Functionally they do the same job, however the guitar and bass pickups are sized and shaped differently to account for the difference in strings.