Is "meeting/room planning" NP-complete?
While my initial instinct was also that the problem should be NP-hard, it looks like it might be close to a problem in P, namely list coloring on graphs whose blocks, or 2-connected components, are cliques. I am not sure to what extent this is helpful, because when you solve the reduced instance all meetings will be filled to capacity, which is an additional constraint on your problem. But if the rest of the argument holds up this might have workarounds, so I will state the partial progress.

If sensible, this reduction does not need to know which day a meeting happens to fall on, so I am using more relaxed notation (dropping the info about which day the meeting was on).

Introduce cap$_m$ vertices $$\{v_m^{(1)}, v_m^{(2)}, \ldots, v_m^{(c_m)}\}$$ for every meeting $m$. Let $M_r$ denote the set of all meetings $m$ that are associated with the constraint $c_r$. the edge set involves a big clique on all vertices involved in representing $m \in M_r$, that is, $$E = \{ (v_p^{(i)},v_q^{(j)}) | p,q \in M_r \}$$
$$\color{red}{\cup \{(v_p^{(i)}, v_q^{(j)})| p, q \text{ are meetings at the same day})\}\text{  [see comment below]} }.$$ 

The color pallette has one color for every person, and the list asssociated with a meeting vertex is the set of all people for whom the meeting is on their preference list. 

A valid list coloring in the graph will fill all meetings to full capacity, and will never assign the same person to similar meetings, and will always give a person a meeting that they wanted to be in.

Some things to note:


The meetings will be filled to capacity, so this is not a completely accurate reduction. The simplest fix is to ask for a partial coloring, but I am not sure about the complexity of this question.  One could make the same copies of every meeting an independent set, and add a complete partite structure across meetings with the same constraint. Now, the color corresponding to a person can be repeated within a meeting. However, I am not completely sure that the graph structure now allows for the Ptime algorithm! The other fix would be to introduce junk colors in the pallette, but then some people may not get assigned to meetings at all.
The reduction works with longer preference lists, too.
Conflicts within the same day can also be handled with this modelling. 


And this paper mentions that list coloring can be resolved in polynomial time when the blocks of the graph are cliques.