In short, no, you cannot use this for all verbs.
The addition of -er/-or to indicate the person or item which performs the verb can be applied to most verbs. As @EdwinAshworth points out in the comments, there are some for which it sounds - at best - questionable ("rainer", "snower", for example), but it's generally a productive suffix which can be widely applied to verbs to indicate somebody/something which performs the verb. Edwin also found an interesting and relevant publication: Combinatorial Restrictions of Agentive Suffixes in English and Serbian, by Vladimir Ž. Jovanović which discusses this among other things. It points out that -er/-or forms can sound strange if there's already an established noun (you wouldn't use spyer/spier in place of spy, for example). But generally speaking, if you need a noun for somebody who verbs and none appears to be established, verb-er or verb-or would probably be accepted by most people as a reasonable coinage.
But the addition of -ee to indicate the person/item the verb is done to isn't nearly so generalisable. Employee certainly isn't the only example (trainee and interviewee, for example, are widely used), but it can't be applied to most verbs.
When I say it can't be applied to most verbs, I mean that with most verbs it sounds unusual and would probably be considered an error by most people, especially (but not exclusively) in formal contexts. I would still expect most people (at least most native speakers of English) to understand your intention (at least roughly). In particular, none of the -ee examples you gave sound natural to me (perhaps transferee, which I would understand as being the recipient of a transfer).
I'd also note that for the two examples where you've provided definitions, I wouldn't interpret them in the same way you have:
merge, merger, mergee (the function is called merge, the one everything is merged into is called merger and the mergees are the ones that go into the merge)
split, splitter, splitee (the function is "split", the original item is the "splitter" and the "splitees" are the results)
for me, I would interpret the merger as being the person/machine/etc that performs the merging. It could perhaps in some contexts indicate the things which go into the merge, but I certainly wouldn't understand it as the result of the merge.
Likewise, the splitter could be the item which splits. But I'd be more likely to understand it as the person/object which causes something to split. I would only understand splittee as being the thing which somebody/something caused to split, and certainly not as the results of something being split.
In general, I'd sum up the relationship between verb, verber, and verbee as: a verbee gets verbed by the verber.
It's possible others may interpret them the same way you did, or another way altogether. But as I said above, neither of those examples sound at all natural to me with -ee so I would avoid using them regardless of what exactly you intend them to mean.
As for how you can tell which verbs you can do this with, unfortunately I don't believe there's any way other than checking a good dictionary to see if a particular form is recognised (or simply knowing from past experience, of course). -ee in particular is not a widely productive suffix, and I wouldn't recommend coining new words with it in serious contexts. It's not too unusual to hear it used innovatively for humorous effect, though.