I hope my answers can be of some use, although they're biased towards a small slice of European history.

Does every historian end up becoming well-versed in the calendars and date-related conventions that was used in the time and area they study? Or are there historians that specialize in calendars and so on that you can ask to do a date-analysis for you?

Historians are generally taught our period's dating conventions early on in our training, during grad school.

As far as the European middle ages are concerned, it's not really that difficult to learn, although it can seem a bit counter-intuitive at first. Medieval people often wrote down dates in terms of 'regnal years' or 'papal years' reckoned according to the length of time that the local ruler or the current pope had been in power. So for instance instead of 12th November 1227 you might see "the twelfth day of November in the twentieth year of the reign of King Henry III". Dates were also often given in relation to feast days: a person would say, for example, that they witnessed an event happening 'on the first Thursday after the Epiphany' - you then have to calculate this date by finding out when that feast day fell in that particular year, and count away from that date.

Fortunately, there are fairly comprehensive reference guides that provide lists of when particular feast days fell on any given year, reigns of each monarch etc (we medievalists have Cheney's Handbook of Dates, for example), so it's not necessary to have all this information memorised. With those kinds of research aids, historians generally don't need to consult specialists to interpret the dates given in a document.

In modern times, we see people writing dates in different, sometimes ambiguous ways (i.e. "01/04/2019" could be the first of April or the fourth of January). Is this a common problem in historical documents too?

I can't say I've encountered this problem myself in my period, nor have I heard of it coming up much in relation to medieval documents. This might be different for other areas of history, but I would assume that in those circumstances, context and topic knowledge would generally clear up any ambiguities.

Is exact dates even important all of the time? Or is it often good enough to know that a document is from a time and place where calendar X was used, and then you move on to examine the contents of the document?

It depends. For the kind of research I do (literary and intellectual history before 1500), scholars tends to deal less in precise dates than in looser decades, half-centuries or sometimes even centuries. On the one hand, this is partly because the rate of cultural and artistic change in an era when manuscripts had to be copied by hand can often be slow when compared with the pace of change enabled by printing. Certain literary forms or philosophical ideas might take several decades to develop and spread across Europe, whereas the same process might take only a few years in the modern era.

On the other hand, this is also the result of the documents themselves. Manuscripts that didn't have some kind of administrative purpose - and this would include everything from scientific treatises, poems, religious texts - often did not have a date written on them. Medievalists can make an educated guess based on different kinds of evidence, e.g. paleography (looking at the handwriting of the scribe and comparing it with known datable trends in style), spellings of certain words, and contextual information about the author's background. Differing interpretations of dating tend to be a big part of the scholarly literature on any given text. These debates are important, but there's usually a lot of other aspects to the texts that other historians will focus on instead.

Of course, for other kinds of historical investigation (e.g. military, political or economic history), precise dating is likely to be much more of a concern.