Is Marginal Revenue not always equal to the price? Marginal Revenue is equal to the price in perfect competition, but MR is also defined as the revenue obtained by selling one extra unit of the good, so how is it not always the case that MR=P ?
Revenue from one unit of a good = Price of the good. 

Or is that definition not true, namely MR should be seen as $\frac{\partial R}{\partial Q}$ (which deosn't really make sense for goods that are discrete things, but let's just admit it does?)
No, marginal revenue is not always equal to the price. It would be if marginal revenue were defined as only the amount paid for the last sold unit, which is equal to the price by definition. However, selling one more unit of a good might have  implications on the market price itself, thus impacting the revenue as a whole.

Consider a monopolist selling a quantity Q1 in a given period at a price of P1. It is a monopolist so it is selling the maximum amount demand can bear at this price. I.e. Qd(P1)=Q1.
Suppose he is now deciding whether or not to sell an additional unit and prices for all goods he sells have to be uniform (i.e. no price discrimination). Since at the current price P1 no additional unit can be sold, the price has to drop to P2 so that: Qd(P2)=Q1+1.

What is the marginal revenue of this decision? It is not equal to P2 since this would ignore the effects on the revenue overall. Instead, the effect is : MR=P2+(P2-P1)*Qd(P1). So on the one hand there is the price of the last unit sold, but on the other hand there is the effect on all other units sold because the price itself has changed.

You are right that a demand curve theoretically speaking is not a smooth function, and shouldn't have a slope but instead be some kind of step-function technically speaking. However if this step function for example implies that every increase of 1 in the price means a decrease of say 2 in quantity demanded, then nothing is lost in making it a linear function of the form Qd=a-2P.

So yes your last definition is correct.