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I wonder if there is an accepted way of describing a fly, let's say a fly new to science? ... I think maybe the way one usually describes a fly is dependent on that specific fly, i.e. with emphasis on the things that distinguish it from other similar flies rather than a general description on the basis of a diptera description template. Why do I ask? If there were an accepted standardized way of describing it, I think, we might have a possibility to put that info into a database and make it searchable regardless of what family it belongs to ... but this is probably an utopia.

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There is no universal template. Descriptions vary the the preference of the author. A normal good description mentions all characters that are known to be variable within the genus (or family), plus anything else that sets the new species apart. Even now, many descriptions are little better than a list of differences between the new species are a previously known one.

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I suspected that, but I also realise that it is probably not an easy task to describe a fly species with all its attributes. I wonder how the taxonomists handle a family like Phoridae with so many species left to "discover" ...

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This is a topical issue.
Recent years have seen some fine examples of how NOT to describe a fly new to science which have promted calls for a 'standard' format for descriptions. This is easier said than done because as you point out, what constitutes a full description varies from group to group and in some cases (eg when comparing with a similar already known species), it may not even be desirable. Surely the golden rule has to be that a description includes enough detail to ensure that confusion with currently known species is impossible and that is is very unlikely with any species that might be described in future. The 'rules' determining what this really means can only be agreed upon amongst specialists with experience of the group of flies concerned.
Allied with the problem of constructing a good description is the problem of designating ia new taxon properly. This is increasingly important as electronic data retrieval is commonplace and poorly construted designation could easily be missed. Some of us think that the Code should be exacting in its requirements for desigations to be valid. If a designation is lacking then it should not be allowed in the Code! (if it hasn't got a comma between the author and date its invalid!)
At the last Diptera Congress in Fukuoka there was enthusiastic discussion over this issue and it became apparent that many journal editors as well as individual dipterists were getting vexed at a perceived tendancy towards poor descriptions and inaccurate designations. Where this leads is anybodies guess but it is up to us as Dipterists to strive for better standards throughout.
This could be a whole thread itself
cheers
Adrian