Am I on the right track, will a linear actuator have enough strength to be able to push down on a common door handle?
Yes, they come in different sizes/strengths/speeds.
How much force does the lever need? You can check it with a pull-gauge (or with weights if it works by pulling down). You'll probably want to use about double the required force to allow for "sticking" and so you don't over-stress the motor by running it at it's maximum.
Alternatives to a linear actuator are a a belt & pulley or worm/screw. These are common ways of converting rotational motion to linear motion. You'd have to do some calculations convert torque into linear force.
It would like to create something that would pull down on a lever/door handle and nothing else.
Typically, you'd replace the lock/latch with a [u]solenoid lock[/u] rather than rigging-up something to turn the handle/lever. [u]Here's are some[/u] for an entry doors.