$\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]

Question

I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'm watching is trying to break the equations down into something grokkable by expanding some of the operators into their identities.

Navier Stokes: $\rho\left[\dfrac{\delta v}{\delta t}+v\cdot\nabla v\right] = \rho g-\nabla p+\mu \nabla^2v$

Mass Continuity: $\rho(\nabla\cdot v)=0$

Convective Acceleration: $v\cdot\nabla v = \left[v_x\dfrac{\delta v_x}{\delta x},v_y\dfrac{\delta v_y}{\delta y},v_z\dfrac{\delta v_z}{\delta z}\right]$

Pressure Gradient: $\nabla p \equiv \left[\dfrac{\delta p}{\delta x},\dfrac{\delta p}{\delta y},\dfrac{\delta p}{\delta z}\right]$

Pressure: $p = k( \rho - \rho_0)$

Diffusion: $\nabla^2v\equiv\left[\nabla^2v_x,\nabla^2v_y,\nabla^2v_z\right]\nabla^2v_x\equiv\dfrac{\delta^2v_x}{\delta x^2} + \dfrac{\delta^2v_y}{\delta y^2} + \dfrac{\delta^2v_z}{\delta z^2}$

Mass Continuity: $\rho(\nabla\cdot v)=0$

Mass Continuity Solution: $\nabla\cdot v = \left(\dfrac{\delta v_x}{\delta x} + \dfrac{\delta v_y}{\delta y} + \dfrac{\delta v_z}{\delta z}\right) = 0$

This to me raises a few questions:

1. $\nabla$ seems, from this, to simply be the derivative of each component of the vector.
2. So $\mathbf{v}\cdot\nabla$ seems to be the product of those derivatives and $\mathbf{v}$
3. But then $\nabla^2$ seems to be the sum of the second derivatives of the components, which doesn't seem right.
4. Finally, there's a $\nabla\cdot$ operator which seems to be the sum of the components of the first derivatives.

So in the absense of an explanation, I'm somewhat confused as to how the $\nabla$, $\cdot \nabla$, $\nabla \cdot$, and $\nabla^2$ operators actually work?

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Comments have both suggested I removed the screenshots and advised me of mistakes on the slides, I also don't want to trust my transposition too much. So, I've converted the bulk of the question to MathJax and included the slides for reference.

Reference Slide 1:

Reference Slide 2:

Answer 1

Treat the $\nabla$ symbol as a vector with the components $(\partial_x,\partial_y,\partial_z)$ and all of this will follow naturally: $$\nabla\phi=(\partial_x\phi,\partial_y\phi,\partial_z\phi)$$ $$\nabla^2\phi=(\nabla\cdot\nabla)\phi=\partial_x^2\phi+\partial_y^2\phi + \partial_z^2\phi$$ $$\nabla\cdot\mathbf{a}=\partial_xa_x+\partial_ya_y+\partial_za_z$$