The Five Cardinals

Question

I've got five cardinals, one without its mate.

My house is made of exactly four-hundred and thirty-five nice solid bricks.

(I'm kidding - they're rather fluffy.)

My length is about two hundred and forty-three, give or take.

My bench has nine legs - really, no tricks!

Mathematically, I have some interesting attributes, too:

a hyperbolic trig function in my middle, a quadratic on my left,

and - once a year - a sphere on my right. Yoo-hoo!

My whole is, in a way, a complicated plane shape who from 2D has left.

Note: just for fun, I made a question with the knowledge and trivia tags - see here for why.

Note: I replaced "chair" with "bench". I think it's a little more accurate this way.

Hint 1:

The specific trig function is the hyperbolic cosine function. It has a very famous application. That application is what that line's clue is driving at.

Hint 2:

The sphere exists every 365 days (366 every four years, of course), and is used to celebrate a certain event that happens with that regularity. It is destroyed at the end (or beginning?) of the event.

Answer 1

Edit - Update

I think the answer is

The United States of America

I've got five cardinals, one without its mate.

Five of the U.S. states have cardinal directions in their name: North Carolina, South Carolina, North Dakota, South Dakota and West Virginia - which is the one without the mate as there is no East Virginia.

Original: Five major self-governing territories - two are located in Micronesia and two in the Caribbean. American Samoa, in Polynesia, does not have a mate.

My house is made of exactly four-hundred and thirty-five nice solid bricks.

The U.S. House of Representatives has 435 voting representatives.

(I'm kidding - they're rather fluffy.)

My length is about two hundred and forty-three, give or take.

This year it is 243 years since 1776, when the United States declared independence

My bench has nine legs - really, no tricks!

The Supreme Court of the U.S. has nine justices on its bench.

Mathematically, I have some interesting attributes, too:

a hyperbolic trig function in my middle,

cosh (as confirmed by OP). I think this refers to the catenary curve, a famous example of which is the Gateway Arch in St. Louis, Missouri.

a quadratic on my left,

A famous example of a parabolic arch is the Golden Gate Bridge in California.

and - once year - a sphere on my right. Yoo-hoo!

This is the Times Square Ball in New York, which drops to celebrate the New Year

My whole is, in a way, a complicated plane shape who from 2D has left.

Geographically, coastlines are often referred to as fractal in nature so may be attributed with a fractal dimension.
Alternatively, because the U.S. is quite a large country, the amount it curves over the surface of the Earth is quite large compared to most other countries so we can say it is definitely not flat (2D).

Original answer

The U.S. House of Representatives