How do we know at what rate the universe is expanding?

They don't use Doppler shift they use Redshift to measure the expansion which is not the same thing (although is often said it is). We know from type 1a supernova that the rate of expansion is in fact increasing. This has been known for about 10 years so you are a bit behind the curve.
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Redshift actually tells us the speed of the galaxies, not their age. We get age from reversing their trajectories, OR, calculations done on the cosmic inflation of the big bang. Both of these match up.

That being said, the universe isn't expanding at the same rate; it's accelerating. We can see this through redshift.
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"How do we know at what rate the universe is expanding?"

We measure size and redshift to objects in different epochs.

"I know scientists used the Doppler Red Shift to determine the age of the universe and that the universe constantly expanding."

NOT Doppler, more complex than that.

"How do we know the universe is constantly expanding at the same rate?"

We know it has a wide history of different rates:
- initial inflation (we are moving at 1000 times c with respect to the CMBR radiation)
- 12 billion years of stagnation (very little difference in redshift between different epochs here, from 4 to 1.1 times c).
- for the last 1.1 billion years we have acceleration of expansion.
.. and note "recession velocity" has no c limit... it is the present "moving away form" the past... not one galaxy moving away from another.