"Quidque velit" seems to mean "et quid velit" in this line from Ovid:
Quos omnes acie postquam Saturnia torva
vidit et ante omnes Ixiona, rursus ab illo
Sisyphon adspiciens “cur hic e fratribus” inquit
“perpetuas patitur poenas, Athamanta superbum
regia dives habet, qui me cum coniuge semper
sprevit?” et exponit causas odiique viaeque,
quidque velit: quod vellet, erat, ne regia Cadmi
staret, et in facinus traherent Athamanta sorores.
Also quidque in Digesta Iustiniani 48.19.6.pr.6:
inquirunt, quid sit, quod allegare principi uelint, quidque quod pro salute ipsius habeant dicere, post quae aut sustinent poenam aut non sustinent.
Actually, I found it difficult to find clear and unambiguous examples. (I think questions with multiple question words (of any form) are fairly rare in Latin, as they are in English, so that could partly explain why it is hard to find examples.) But
in general, it is possible for -que to be used as a coordinating conjunction even in cases where it creates homophony with another word ending in -que. So I think "Quis es quidque facis?" could be used with the sense "Who are you and what are you doing?"
When attached to a relative pronoun, -que just has its normal usual sense of "and", e.g.
nam quanti refert te ei nec recte dicere
qui nihili faciat quique infitias non eat? = [et qui infitias non eat?]
(Plautus, Pseudolus 1086)
praedicationem non me praeterit, si, quem desideramus
agricolam quemque describemus = [et quem describemus]
(Columella, De Re Rustica 1.pr.28.6)
I think examples of relative pronoun + coordinating -que are not that rare.