is it safe to tell them my current salary and that my expected salary
is the listed salary?
I wouldn't but it could be safe. Some places would pounce on that if you are really underpaid so much, that they could make a killing off of representing you as companies may pay quite a mark-up that you'd be OK given you get a bit of a bump up and the company takes a major cut of the profits here. Think about this way: If the company can bill you out for $200/hour and give you $40/hour, they make a mighty nice profit off of you in contrast to someone that bills out for the same rate but takes $140/hour instead.
Or should I just mention the expected salary?
Yes, just mention the expected salary as that is something to note in most employment agreements. I agree to be compensated at $x/hour or $y/year generally. This is useful in positioning where you are generally. Are you on the higher end, the lower end or something else.
Or not mention anything at all?
If you want to appear difficult, you could go this route. The company wants to know what kind of rates you expect so that you aren't sent on jobs you'd turn down because they don't compensate adequately. If you make $60,000/year and a job comes up that pays $15,000/year there is little chance you'd want that job and though this is why they need to know an expected salary.
Can a recruitment agency offer me less than what is mentioned on the
website once I mention my current salary?
Yes, it is possible to be offered less than what is on the website as they could claim that the client isn't willing to pay what they thought given your background. Some jobs may advertise $80-100K and yet if the person that makes it through the interview isn't quite thought of that highly, a company may only be willing to offer $65K in which case you then have to decide what are you thinking. The other possibility is that when you mention your current salary that the recruitment agency send you out the door given the pay difference between where you are and where you want to be. Some places may well try to talk you down a bit since this is such a jump. There can also be the question of how well could you handle the potential job in responsibilities if it pays so much more that they may expect more? Not guaranteed but it is possible.
While I have had some places where I worked that had a bit of a pay cut, it is a tricky thing to present in a way that doesn't place one in a bit of a corner. In taking the low rate previously, this opens the door to wonder, "Why wouldn't you do that again, hmmm? Maybe we could work something out..." kind of thinking that isn't likely where you want to go.