How to derive the formula for phase difference $\Delta \Phi = \frac{2\pi}{\lambda}\Delta x$?

How can I derive the formula $$\Delta \Phi = \frac{2\pi}{\lambda}\Delta x$$ for calculating the phase difference?

On a relative note, why does the particle velocity have an upwards direction when the wave travels below the x-axis? Or, what exactly is particle velocity?

$$y_m \sin kx_1 = y_m \sin k(x_1+ \lambda) = y_m \sin (kx_1 + k\lambda)\;.$$ A sine function repeats itself after an increment of $2\pi$. So, \begin{align}k\lambda &= 2\pi \\ \implies k &= \frac{2\pi}{\lambda}\end{align} which is wave-number, the spatial rate of change of phase or wave-cycle.

Suppose two waves having same phase initially, travel different distances $x$ nad $x + \Delta x$ and interfere at a certain point. Then the two waves are written as \begin{align}y_{\mathrm I} &= \gamma_1 \sin(kx - \omega t) \\ y_\mathrm{II}&= \gamma_2 \sin(k(x + \Delta x) - \omega t) \\ & = \gamma_2 \sin(kx - \omega t + \delta) \\ \implies \delta &\equiv \text{phase difference}\\& = k\Delta x\\& = \frac{2\pi}{\lambda} \Delta x\;.\end{align}

[Sorry, for the abuse of notations.]

If the path difference is $\lambda$ then what would be the phase difference?

Its $2 \pi$. Now If path difference is $\Delta x$ the phase difference would, what you are searching for :)

Phase difference is nothing but average wave number multiplied by path difference $\dfrac{2\pi x}{\lambda}$ where $x$ is distance between two particles $x$ can further be written as: velocity $\times$ time

That is Phase difference $= 2 \pi v t/\lambda = 2\pi n t$ Where $n$ is frequency

The above formula is more of an definition rather than a derived formula. Without getting too mathematical, a wave, among other things, is defined by its phase. A wave can be described by any combination of wave-functions, i.e, functions that solve the wave equation:

$u_{xx} - (1/c^2)u_{tt} = 0$

Every function of the form: $f(kr - \omega t + \phi)$ can solve the wave equation. The argument of the wave function is the wave's phase ($kr - \omega t + \phi$). It's usually comprised of a temporal phase ($\omega t$), a spatial phase ($kr$), and and an initial phase ($\phi$). The spatial phase term is $kr$, where k is a vector with magnitude $2\pi / \lambda$. It's also called the 'wave number', or 'spatial frequency'.

It now follows that any displacement $\Delta X$ will result in a phase difference $\Delta \Phi$.