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This is discussed, e.g., in P5, Q2.a

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For a non-uniform body, is M (total mass) given by the integral of the volume mass density function ρ(x,y,z)?

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For a non-uniform body, is M (total mass) given by the integral of the volume mass density function ρ(x,y,z)?

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The Lagrangian L is an explicit function of many variables, thus it makes sense to consider partial derivatives.   In the systems considered in our module, L  it is an implicit (never explicit) function of time, via the other  (explicit) variables. And a total derivative of time shows up in the E-L equations. But this does not answer the question.
E-L eqs can be obtained via a rather elaborate derivation in constrained systems, using the so called "principle of virtual work", which is out of the scope of the module. An alternative, much more elegant derivation of the E-L eqs. is the one that makes use of the the action. The action is a functional, i.e. a function of functions. We will see in later lectures, that minimizing the action naturally leads to the E-L equations.  If you want to have more information, please have a look at Taylor and Landau, two books which are in the module reading lists.  I hope this helps.

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The nabla operator  is a "formal" vector, whose components are partial derivatives with respect to suitable variables (e.g. cartesian). If applied as such, nabla acts as a gradient of a scalar function (of those variables). If nabla is combined with dot and cross products, it produces, respectively, the divergence operator and the curl operator. See Lecture 3 (L3)- Supplementary material- Slide 5.

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What is a operator nabla?

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What is the motivation behind using partial derivatives instead of a total derivative when deriving the differential equations to model motion?

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In general, the formula to obtain an arc ds of a circumference of a radius R, given the underlying angle d\phi, is
ds = R d\phi     (this is a scalar equation)
This is a general formula that it is everywhere in maths and physics.
In this specific case (the slide you posted), the modulus of the vector |\Delta r^hat|  plays a role analogue to the ds above, i.e. it will be the length of the arc of a circumference of radius |\r^hat| with the underlying angle \Delta_\phi.
You will then have
|\Delta r^hat|  =  |\r^hat| \Delta_\phi  (this is a scalar equation)

Note, however that |\r^hat|  =1 as \r^hat is a unit vector.
Hence
|\Delta r^hat|  =  \Delta _\phi

However, we are interested in calculating  \Delta r^hat, i.e   a vector (not only in calculating its modulus).  For very small \Delta r^hat,  \Delta r^hat tends to be directed as the other unit vector, \phi^hat.
Then I can write

\Delta r^hat  =  \Delta_phi   \phi^hat   (this is a vector equation)

At this point I can divide both member by a finite \Delta t.
Then I take the limit \Delta t -> 0.
On the lhs this gives me the time derivative of the unit vector r^hat (by definition of derivative).
On the rhs this gives me the time derivative of the angle \phi, i.e. \dot{\phi}.  Times the unit vector \phi^hat

I hope this helps.

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Calculating the time derivative of polar coordinate unit vectors

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Calculating the time derivative of polar coordinate unit vectors

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How to do you get the circled equation from the information in the diagram and the phi unit vector. Also

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Very good question. Yes, it is a perfectly working strategy: You transform the human body in a set of centres of mass, each with a sub-mass associated. Then you find the centre of mass of this (at this point discrete) distribution of points (centres of mass). However, the human body has iso-density parts (= parts with the same density)which are spread out on the whole body (e.g. the bones, the muscles, etc). So you need to know the iso-densities parts...Note, in passing, that the human body is not a rigid body, i.e. given two points of the human body, their distance can change in time (soft matter, the arms and legs can move, etc).

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What other methods you refer to? We essentially use Calculus. As velocity and acceleration are derivatives of the position vector, it is natural to set up equations with derivatives, and then use integrals to solve them. Solving them means to find functions of time (trajectories) which describe the motion at any time.

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For a body of mass in 3D with uneven mass density. (like a human body), when calculating the centre of mass would you separate areas of certain density withing the object, work out centre of mass of each individually and then combine as though all the individual parts and now point particles as shown in the original centre of mass equation. Human body isn't best example because of its complexity, however would the method work in theory.

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Why use a differential equation over other methods

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Your weekend?

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How do you solve a Euler-Lagrange equation for a lagrangian with multiple variables. E.g:

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what is the tensor of inertia of a body?