does a design require a designer?

No. Snowflakes "design" themselves into an infinity of beautiful, near perfectly symmetrical, patterns based on the random freezing of water and can differ according to how cold the temperature is. There is no "snow designer". The whole "does a car build itself?" analogy is pointless.

Well lets take that snowflake and examine that. In order for a pattern, there must be certain elements that come together in order for the patterned snowflake to come into existence. And what is the engineering that took place for the elements to work together? And where did the knowledge for that engineering come from? And where did that knowledge to know that engineering was needed to design the elements to form a pattern? Every design has to start with a vision. Some Source of Intelligence is needed to visualize that design.

No one can really prove whether the universe was designed or whether it came about through natural selection & chance, though I believe in the latter.

If it was designed, so much has changed since then few of the original designs exist anymore... so if it was designed, why didn't the designer (if He is so all-knowing and wise) create a perfect universe instead of one that needed to evolve to be better?

I'm a programmer. I design information systems. They DO not just come into existance. You have to have a toolset, and environment to create the finished product, and then you have to use logic to put everything together into something that fits the requirements that you have. Without those basic elements, it is impossible for something to be designed.

Does a building have a builder? Or did the building build itself?

Of course not!

Yes there is a designer! There is a Creator! And it is God.

Hebrews 3:4

(For every house is built by someone, but the builder of all things is God.)

Design does need a designer, nature does not.

Does a house have a designer, or does some one just haphazardly build it and hope that everything turns out okay?

The universe wasn't designed, neither do I think it's a conscious entity capable of self-design. It's a result of cosmic evolution.

By definition, yes.

an accident, no. But a design... yes.

You can't prove that that is the case. It is at best a heuristic.