Using the standard IS-LM framework also used in Khan Academy videos the demand for loanable funds can be expressed as (Fields & Hart 2003):
$$LF^d_t = i_{t+1}(r) + (g_{t+1} - t_{t+1}) $$
where $i$ is the investment spending, $g$ government spending and $t$ lump sum taxes. The subscripts are there for future because agents are forward looking you can ignore them if you want to integrate them into undergraduate level analysis shown in Khan Academy. The supply of LF can be described by:
$$LF^s_t = s_{t} + \left(\frac{M^s_{t}}{P_{t}} - \frac{M^d_{t}(P,y^*,r)}{P_{t}} \right)$$
Where $s$ is the saving level $M^s$ is money supply and $M^d$ money demand and $P$ price level.
What effect would a change in the demand of money have on the supply and demand of loanable funds
Positive or negative changes in money demand do not affect loanable funds demand directly we are moving along the loanable funds demand curve (as price of funds changes) but there is no shift in demand for loanable funds (the quantity demanded will be changed though but that is depended on shift in supply).
Positive (negative) change in $M^d$ will cause supply of LF shift to the left (right). So the new intersection will be with less (more) equilibrium quantity of loanable funds and higher (lower) interest rate.
What effect would a change in the money supply have on the supply and demand of loanable funds?
Again no shift in $LF^d$ we are still moving up and down along $LF^d$ depending on what happens to $LF^s$.
Positive (negative) money supply change would shift $LF^s$ to the right (left) leading lower (higher) $r$ and higher (lower) equilibrium quantity of LF.
Why does the money market use NOMINAL interest rates as a measure of price, yet the LF market uses REAL interest rates as a measure of price?
Its a stylistic choice, you can use real interest rate in money market as well. For example, this random lecture note about money market uses the real rate. I also seen different textbooks introducing this market with real rate although admittedly its more rare than having nominal rate there. Both real and nominal interest rates are connected through Fisher equation $r\approx i -\pi$ with some small adjustment you can rewrite any model that has real interest rate in terms of nominal interest rate and vice versa. I suppose most textbooks choose to use nominal interest rates because money market is used to explain central bank's intervention and central banks can only directly set nominal interest rates. I suppose that could be the reason for the stylistic choice, there are no surveys of textbook authors on this issue.
