Electronic – Voltage, Current, Angular Velocity, and the effect they have on Torque

A gyro measures angular velocity, and is relatively insensitive to other movement. You would need to differentiate to get angular acceleration. An accelerometer measures acceleration. Period. It will respond to reorientation with respect to gravity (as gravity is an acceleration). It will respond to linear acceleration. If offset from the axis of rotation, it will respond to centripetal acceleration, which will be related to angular velocity (though if motion is sinusoidal, you will need to keep track of tangential accelerations).

If you can orient the accelerometer to be right on the axis of rotation, you can easily track the orientation with respect to gravity (that is, Z), but not so easily in the xy plane (if that is the earth plane in your system)

Your situation may be easier to use some sort of absolute or relative encoder, which will give you angular position without worrying about drifts and offsets associated with integration.

With your improved description, you might want to die toy measure torque! http://www.omega.com/googlebase/product.html?pn=TQ513-514-FOOTMOUNT&gclid=Cj0KEQiAtZWkBRC9ibSfhoKEyLYBEiQA5fDxkcDLSo3_jC7_lsm5amhIy02A4MfjEriBgxXRaW5BxcYaAgOy8P8HAQ

This battery should be able to
power a pic 24 continuously at 0.63 mW, should be able to
keep a fingerprint reader on continuously at 429 mW, and
keep a wifi module in standby mode continuously at 3.96 mW.
I am also considering attaching a dc motor to the deadbolt latch and driving it with an h-bridge. This would also be powered by the battery at 1 W, but this isn't continuous and would last for about 0.5s.

You say you want a continuous mean power output of
0.63 + 429 + 3.96 mW + 1W very occasionally
= 429mW + irrelevant.

In a day that's about 10 Watt.hours

So a user inputting 3 Watts continuously via a door handle would need to do so
10/3 = 3 + hours per day.
Or 30 W for 20 minutes - that's a significant exercise level.
Or 300 Watts for 2 minutes - most people could not do that.

ie you can easily see with such simple calculations that what you have asked for is unrealistic AND that you do not need most or the power you requested. It seems extremely unlikely that you need a continuously powered card reader or that it would need siuch a high poer level in standby mode.

Try again.
5 mW to 10 mW average should be enough.
Over 24 hours, 10 mW =~ 0.25 W.h
0.24 W for 1 hour
2.4 W for 6 minutes
24 W for 36 seconds.
If you can obtain that in 10 sessions it's 24W x 3.6 seconds x 10
Or 2.4W x 3.6 seconds x 100

The last figure sounds like what you might easily be able to obtain from opening a door without it bein too too obvious. In practice you'd want several times more to allowfor inefficiencies and practicalities.

Is 100 operations a day too many - that will depend on application.
But the above gives you a basis for a realistic calculation using assumptions that suit you.

Once you have refined the spec we can talk about power generation.

At 100% efficiency

Watt.seconds ~= kg_force x metres travelled x 10.

This could be derived from opening a door, pulling on a lever or eg standing on a step (that moves under applied weight to power an alternator).