Will the monopolist inevitably increase the price if demand grows? Supposing that demand for a product in a monopolised market has grown, is it possible that equilibrium price will go down?

In most cases we simply draw curves D, MR, MC and make conclusions. I usually deal with linear demand and constant marginal costs, so nothing interesting. 

It is naturally to say that the monopolist will increase the price to gain profit, even for inferior goods this seems true.
In this post, we find that the optimal monopoly price can be written as

$$P^* =  \frac {|\eta|}{|\eta|-1} MC $$

where $\eta$ is the elasticity of demand with respect to price, and so optimal price will necessarily be at a point where $\eta$ is higher than unity in absolute terms.

From this one can deduce some unambiguous and some ambiguous scenarios when the demand schedule shifts outwards, e.g.

SCENARIO A: constant price elasticity of demand, convex costs: $\implies$ Price and quantity will increase.

SCENARIO B: constant price elasticity of demand, linear costs: $\implies$ Price will stay put, quantity will increase.

The floor is yours. 

PS: Note that to have constant price elasticity of demand, the demand function will necessarily be non-linear.