Is the money market equilibrium interest rate the same as the IS-LM equilibrium rate? If IS-LM equilibrium interest rate is, let's say, 5%, does it also mean that the money demand and supply equilibrium interest rate would also be 5%?
Yes. The LM curve is a combination of all such points where money market is in equilibrium. You can see this from the picture below I took from Blanchard et al. Macroeconomics: a European perspective.

The IS-LM equilibrium occurs when IS-LM curve intersect as seen on picture below (again taken from Blanchard et al.)

No matter where IS-LM points intersect there will be a direct mapping from any point along LM curve (as shown in first picture) directly onto the money market. Hence any equilibrium IS-LM interest rate will be also money market equilibrium interest rate (as correctly pointed out by Giskard in comments these are the same interest rate). Hence, in the standard textbook exposition of the model these will be always same.