TL;DR
Intuition is one of the things that distinguishes humans from machines. We should use it, especially to guess what we would like to prove. However, then, we ought to prove our claims.
Introduction:
I would like to start with dividing mathematical problems into two groups:
- straightforward problems, where mechanical work is enough,
- complex problems, where there has to be something more in the proof.
Intuition comes with a cost of introducing errors, and while dealing with problems from the first category we should avoid it unless it outperforms the cost of manipulating overcomplicated expressions. However, while solving a problem from the second group we have to use intuition, there is no other way (well, we could ask someone or do a brute-force search, but in most cases this is infeasible).
Quite long digression about automated theorem proving:
There is a whole research area called automated theorem proving which tries to make computers prove theorems. It is worth noting that there are results, like Robbins conjecture which eluded humans, but were proved by machines. However, there is a vast number of results that are inaccessible to machines (existence of such problems is implied, for example, by undecidability of the halting problem).
I would like to emphasize one aspect which makes theorem proving hard. It is not exactly right, because right now the use of many various heuristics allows computers to solve a multitude of practical problems. Nevertheless, the core issue did not disappear and shows itself in more complicated cases.
The aforementioned issue is that, if the proof behaves in some non-monotone way, the computers are unable to guess the appropriate middle formula that would connect the premises and the thesis. For example, the proofs are usually of the form
$$\textit{premises} \to \textit{something} \to \textit{thesis}.$$
The proof would be straightforward if one could deduce something going forward from premises or going backward from thesis (perhaps doing both). However, it might happen that the inference looks like
$$\textit{premises} \to \ldots \to p \land (p \to q) \to q \to \ldots \to \textit{thesis}.$$
Observe that there is a some form of non-monotonicity between $p$ and $(p \to q)$, namely, the weaker $p$ is, the easier it is to prove it from the premises, but the harder it is to prove $(p \to q)$; the stronger $p$ is, the easier is for $(p \to q)$ but the more difficult for $p$. Independent of whether we go forwards from premises or backwards from thesis, we might have to check an impossible number of paths before we decide if the theorem is true (produce a proof) or not (produce a counterexample).
On the other hand, if we had to prove two theorems, first $\textit{premises} \to p \land (p \to q)$ and the second $p \land (p \to q) \to \textit{thesis}$, then it would be much easier (provided there's no similar situation inside the subparts).
Why it matters:
There is a professor who said, that it is the modus ponens / rule of detachment where the genius in the proof hides.
The hardness is to know the middle lemma that makes the proof work. In other words, the difficulty lies in what we try to prove, all the rest is straightforward (of course, that does not mean easy). A great example of this phenomenon are theorems which we have to strengthen to make them easier prove (see here for some concrete examples).
However, it is the intuition that tells us what to prove. It is a necessary component of any research that deals with more that just a straightforward tasks. It is the best leader we have – we don't have anything else.
Conclusion:
I think it is great to use intuition and to encourage the students to use it.
Every time I can, I encourage them to train their intuition.
Nevertheless, there is an important part: we have to actually prove our results. Intuition might be great, but it will not substitute for a proof. There is nothing wrong with having bad intuition if we learn from our mistakes. Once we will start forming a proof, we gain more and more understanding that will correct our intuition.
Some suggested that they alternate between trying to prove the hypothesis and finding a counterexample. From my perspective both actions deepen our understanding and indirectly correct our intuitions.
Intuition should be trained and that can't be done without using it. Perhaps our solution will be inferior or suboptimal, but thanks to that, next time our approach we will be better. On the other hand, if we were to keep using the same techniques again and again, we won't advance as much.
To answer your second question, I don't believe there is any general way to determine how much intuition do you need in any particular domain. I would say experience is the best you can do, but even then the new generations might have a whole different view, i.e. new tools and techniques might allow for new points of view that might turn everything upside-down. In fact, if you understand some result deep enough, it is intuitive (despite how non-intuitive it might seem to others).
I hope this helps $\ddot\smile$