Ok, I can understand the language issue and your most recent post points up that you do have some issues that need to be addressed.
First off Timing. To properly check timing in the MOST accurate manner you'll need some sized but unprimed casings. This will allow the casings to assist in registering the cylinder to the extractor star. As for why, at some periods the fit of the extractor star to the cylinder was a bit "loose" and casings in the cylinder were the final device to register the extractor star to the cylinder. If your not too fussy about duplicating a cylinder loaded with fresh ammunition you can use fired casings as a substitute but this will improve the registration of the two parts by a small amount. Azoom brand snap caps are also a suitable alternate and I've found their diameter is basically the same as a fresh factory cartridge. If you can get Azoom snap caps in Sweden they are good to have on hand.
Now to check the timing using a method that is considered proper by the largest group of users and gunsmiths. Note, there is some debate on the proper method so what I am presenting is the consensus of what is considered correct. The way to do this is to pull the trigger VERY SLOWLY in Double action and listen for the cylinder stop to crop into it's notch before the hammer falls. If your trigger control is good enough you should be able to drop the cylinder stop in every notch without dropping the hammer once. Now, due to the design of the lockwork in a S&W it is typical for a left handed trigger pull to cause a failure in this test on some chamber positions. The cause is the trigger cocking slightly on it's pivot pin and "throwing" the hand away from the drive pawl. However, with a right handed trigger pull the revolver should pass this test for every chamber position. Failure to pass this test is called a "failure to carry up". Correction for this condition varies depending on how bad it is.
If only one or two chambers are effected the pawl on the extractor can be peened to widen it enough to allow the carry up to be corrected. If every chamber fails to carry up a wider hand is usually fitted.
Now, one note about Timing in terms of Combat Usage. That is that it's obvious that if the trigger is pulled quickly a failure to carry up won't be a problem because the rotating mass of the cylinder will carry it into lock. So, if you are engaging in action shooting a very minor miss on timing may not be worth bothering with. Mainly it's a concern for those who "stage" there triggers to achieve the best accuracy. It's also why S&W advises against staging the trigger in the owners manual, the simple truth is that these were designed and intended to be combat revolvers from the start. Because of this, these revolvers have clearances in the various features that put the priority on Function instead of perfect accuracy.
Now, the good news is that a very good and attentive gunsmith can perfect the timing on a S&W revolver, however it's not as easy as it seems. The problem is that when you try and fit the lockwork on these guns tight enough to "perfect" the left handed carry up you can easily set up a Hand/Cylinder Stop bind. Typically this results in a trigger pull that gets noticeably heavier at the hammer release or stalls completely. Basically, it's a VERY narrow ledge in terms of function. Personally, I don't worry about left handed carry up issues and will allow a very minor miss on a right handed trigger pull. However, I don't stage my trigger in double action, if I'm shooting for accuracy I do it in single action.
Now, about your single action trigger pull. I'll tell you right off that your trigger pull is NOT Factory. In fact it's so far from being factory that I would consider it unsafe. I also think that it's very likely that someone other than the factory has been stoning the sear surface on the trigger in your revolver. I don't know who you purchased that revolver from but I would be having a long discussion with them about selling you a revolver that has been "tinkered with". Chances are very good that you now need both a new hammer and trigger and the cost for that should be born by whoever sold you that revolver.
Now for your sighting issues. Now that the misunderstanding has been cleared up it now seems that what you need is a taller front sight. Take a look at the sight base on your barrel above the muzzle. I expect that you'll see a small hole in the font of the sight base. If so, this means that your 625 has the interchangeable front sight system which is spring loaded. Press the sight blade towards the rear of the revolver and then lift the front up and out of the nest. Now take a look at the reverse dovetail base of your sight and commit that shape to memory. Following is a link to a new front sight blade that I believe may solve your sighting issues, if not you can also use your current sight as a pattern to have a machinist make up a custom front sight of any height you need.
Product: Gold Bead Interchangeable Patridge Front Sight .300 High
PS; I realise that my post may not be the news you wanted to hear. Unfortunately I suspect that somewhere on your side of the Atlantic someone has done some "tinkering" to make it a IPSC "special edition" and they didn't know what they were doing. Basically, they took a perfectly good revolver and butchered it with a hack job. As for why I believe this, it's your single action trigger pull. Quite simply I don't know of any method to reduce the single action trigger on a S&W revolver this much except by removing the trigger and stoning the single action sear to an "assist angle". Since it's obvious the trigger has been modified to a non factory condition it's also possible that your timing issues are also a result of tinkering. Hopefully, whoever sold you that gun will stand behind it and correct the mistakes that have been made here. If not, I would suggest that you look into purchasing the S&W Shop Manual by Jerry Kuhnhausen, some tools, and learning how to do your own gunsmithing. It's really not that difficult to learn, basically it's a matter of attention to detail and being meticulous.
PSS; if you do need to replace the hammer and trigger I would suggest that you change the lockwork over to MIM parts. Doing this will save you from having to "fit" the hammer and trigger together because the MIM parts are normally drop in parts.