How is rotor airfoil analysis different than fixed wing airfoil analysis?
You are correct on many fronts.  Rotor blades see a varied load along their length, resulting in twisting, bending, and lagging (e.g., the blade being bent against its direction of rotation) loads.  It's a mess.  And, yes, all those effects feed back into blade aerodynamics, but that's a much more complicated answer that you might be looking for.

What you are describing is the foundation of blade element theory.

Let's consider a helicopter rotor, independent of the helicopter fuselage, and look at one annular ring of it while the rotor is in hover (so we have an axisymmetric flow field):



If we look at that annular ring, it would "cut" through a section of the rotor, which, in 2D, would look something like an airfoil.  If we drew a bunch of vectors on it to represent the forces acting on the airfoil and the direction of the relative wind, it does look a heck of a lot like our 2D airfoil free body diagram from fixed-wing aerodynamics.  And, truly, blade element theory (BET) considers it as such.



In BET, elements of the blade (such as the 2D airfoil above) operate without impact on the other blade elements.  If you can determine the inflow at a particular blade section, you can determine the sectional lift and drag, and, integrating those properties across the blade, you can determine the thrust production and torque requirements that correspond to a particular operating condition.  That said, while most of the labels on that diagram are familiar from fixed-wing flight, there are a few that should be different, namely: the induced velocity, inflow angle, and tangential velocity.

As you stated, the horizontal speed of the blade is going to change based on where you are on the blade, radially speaking--which is absolutely correct.  If we look at a distribution of elemental velocities around the rotor, it's going to vary as shown here, in a plot from Leishman's Principles of Helicopter Aerodynamics (which has a great description of how to implement all of this, as well as some basic corrections for going into some rudimentary higher-order approximations).  Note the hover case on the left hand side, where the velocities are axisymmetric, versus the forward flight case on the left hand side, where the velocities are asymmetric.  That second case is a whole 'nother can of worms, and we'll get there in a minute.  Stick with the hover case for now.



So knowing your radial location can get you the element's velocity, and knowing your climb rate and help you get how fast flow is moving vertically through the rotor, but flow has to be moving vertically through the rotor even when the aircraft's not climbing -- else our analogy of a rotor being a big fan (which it is) doesn't correspond with reality.  Hence, there's some inflow speed, and it makes sense that that inflow speed is correlated to the amount of thrust being produced by a given element of the rotor...and this is the crux of the problem for us, since it has to capacity to dramatically change the angle of attack of the blade section (and thus our lift and drag predictions).  So: how do we estimate the inflow velocity?

A simple approximation uses momentum theory (i.e., a quasi-1D analysis on fluid moving through the rotor disk, treating the rotor as a device that simply adds momentum to the flow) to determine the inflow on the rotor along that annular section, assuming that the inflow is axisymmetric.  For what it's worth, it's a simple way to capture inflow velocity variation (and thus angle of attack variation as well) across the rotor disk, which is something that momentum theory cannot do alone, and it's decent at what it does.  It has been the de-facto standard of the wind energy industry for awhile as more complex CFD computations have caught up to the industry's analysis needs and desires.

However, when flow is no longer axisymmetric, we get into a big mess of approximations to determine what that inflow does look like, since the wake will start to interact with the inflow as well and that, somehow, needs to be accounted for.  It is here that we start to depart from analytical expressions and look to computational models using free-wake theory or higher-order models.  Where fixed-wing models advance in similarity by modeling the wing's wake, viscous effects, etc, rotary wing models model a rotating wake that is being convected and recirculated by both rotor effects and the freestream.

That said, the expressions for a rotor in hover are relatively easy to code up, depending on how much fidelity you want (e.g., how much you know about the airfoils used on the blade as well as how the blade tapers and twists along its length).  Googling around, I found a MATLAB code that might be useful, again, depending on your problem: https://www.mathworks.com/matlabcentral/fileexchange/21994-analysis-of-a-rotor-blade-system-using-blade-element-momentum-theory