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The play (front to back) is called End Shake and can usually be taken out or minimized with the use of shim washers (.001" and .002" stainless steel sold by Powers Custom).

Another method is to have Yoke tube "stretched" with a gun smithing tool that literally increases the length a few thousandths. Either way is proper but the shim method is easy, cheap and something you can do yourself.

As far as timing goes, the Cylinder should lock up prior to the Hammer falling when the Trigger is slowly pulled. If not, the firing pin could actually miss the Primer when slow fired in double action mode. That too is usually an easy fix with the installation of an over sized Hand. They are a bit wider and they move the Cylinder a bit more into position. They will usually need minor fitting.

Please note that some amount of play is by design and perfectly normal. If everything were too tight your Revolver would jam up. If you pull the Trigger slowly and the Cylinder does lock up prior to the Hammer falling - you are OK. If you do a search here on how to measure End Shake you can do that yourself. Again, a little bit is OK.