Beneficiaries of an action ending with the "-ee" suffix
The suffix -ee comes from the French past participle suffix -é(e). There is a relatively short list of English words ending with -ee, the vast majority of which are French or Latin in origin. I suppose killee and givee sound wrong because they’re of Anglo-Saxon origin, not Romantic, so the -ee suffix is less natural.
In general, I would avoid using -ee to coin new words, and further avoid using -ee words other than the most common, such as employee, attendee, detainee, &c. The silliest-sounding -ee words seem to be those using native English roots, where -ed is better:
*The mugger mugs the muggee.
The mugger mugs the mugged.
All the comments that follow appear to me to be true of my English. No guarantee that they apply to others'. Comments welcome. Thinking about examples that I feel more and less happy with, I'd say that there are some prosodic constraints on -ee.
In the best examples, -ee attaches to a root that has (i) two syllables, (ii) has, or can have, primary on the first syllable, (iii) has no long vowel in the second syllable. Your examples assignee, employee, and refugee all fit this description, as do nominee, advisee, and conferee.
Note that, in the last two, stress shifts from its normal locus (adVISE, conFER) to the first (ADvisEE, CONferEE). This is why (ii) says "can have", not just "has". Some prefixes seem less able to accommodate this stress shift: forgetee is completely impossible for me, and impugnee, repressee are pretty iffy.
Nondisyllabic roots are (un)acceptable depending, again, in part at least, on prosodic factors. Two unstressed syllables before ee is problematic: witness such contrasts as convertee (imaginable) ~ controvertee (not), or preposee (imaginable) ~ presupposee (not). Similarly, one long syllable is also uncomfortable, as in your massagee. But examinee and eliminee are both fine for me, with the extra syllable occurring before the main stress.
When it comes to monosyllabic roots, if the root has a long vowel, I find the examples slightly better than if the vowel is short. I'm fine with slayee (a Buffy example), callee (cited above), freeee (it's about time we had a quadruple vowel in English), but not so happy with givee, dropee, kickee. To make the latter three vaguely acceptable, I find myself inserting a glottal stop between root and suffix (kick'ee, etc.).
That said, though, phonology is clearly not the whole story. I'm fine with confutee but unhappy with confusee, unhappy with refutee but fine, of course, with refugee. Or maybe I'm just tired...
Unfortunately English tends to be a little bit random in the way rules such as this one are followed, partly as it often tends to be based on the etymology of the word - i.e. the language that the word has been borrowed from.
That said, you could get away with adding "ee" and "er" to pretty much any verb and the person hearing it, if given enough context, would be able to understand you. For instance, there is a memorable exchange in the TV sitcom Friends:
Monica: Oh man, they think they are so slick messing with us! But see they don't know that we know that they know! So…
Chandler: Ahh yes, the messers become the messees
"Messer" meaning people who mess and "Messee" meaning people who are messed
Neither of these words are in general use, but the language works because enough context is given (the verb "mess" had been used in the previous line"), the character is known as a comedian and the words themselves are emphasised when spoken.
However, for anyone who is not themselves thoroughly fluent in English my advice would be simple: only use this construction if you have heard or seen the specific word used multiple times elsewhere. Otherwise you may end sounding awkward or pretentious.