What's the English cognate with German "Stick" and Dutch "stik"?
Yes, there is a cognate: stick (as in stick out or stick a needle in someone).
The German and Dutch words (as well as Scandinavian equivalents like Danish kvælstof and Swedish kväve) are derived from verbs meaning ‘smother, stifle, strangle’, since nitrogen cannot sustain fire, but ‘smothers’ it. So a literal calque would be ‘smother element’.
The German and Dutch verbs are cognates; I'll just focus on the German one here. The verb in German is ersticken, which is an er- extension to the base verb stecken (not as one might expect sticken, which is a different, though related, verb). Stecken means ‘stick’ in various different senses related to putting something in some place or into a hole (sticking a nail in a wall, etc.) or fastening something (making it stick); it is in origin the causative of stechen, which means ‘be sharp, sting, prick’. So if a nurse injects a needle into your arm, the needle sticht you, the nurse sticht you, and the nurse steckt the needle into your arm.
Etymologically, all these verbs are of course related, in quite complex ways. Stecken in Modern German represents a merger in Middle High German of two verbs that were in Old High German distinct (though very similar), which Duden quotes as stecchen ‘stick/adhere to, fasten to’ and stecchēn ‘stick into, pierce’. Of these, the latter is the original causative (Old High German verbs in -ēn generally come from Proto-Germanic *-janą, which was the standard causative ending in the infinitive). Of course, it’s not really a true causative at all (a true causative of ‘be sharp’ would be ‘make sharp’ or ‘sharpen’), but the causative in Germanic was quite a broad concept, and there are many morphological causatives that aren’t semantical causatives.
I haven’t seen any conclusive explanations on the development from sticking/pricking/sharpness to strangling or suffocating. In fact the DWDS etymology for the verb ersticken pronounces itself uncertain:
Der semantische Zusammenhang ist undurchsichtig. Ist für ersticken von ‘durch mehrere Stiche töten’ (vgl. mhd. si lāgen ersticket und verdorben) mit nachfolgender Bedeutungsentwicklung zu ‘(an Luftmangel) sterben, (durch Entziehen der Luft) töten’ (wie ¹DWB 10, 2, 2, 2742 vermutet) auszugehen? Oder führt ‘hineinstecken, stopfen’ zu der Vorstellung ‘durch (Voll)stopfen den Hauch, den Atem benehmen’ (vgl. mhd. ersticken und erworgen begunde er an dem beine), also ‘mit dem Atem steckenbleiben und den Geist aufgeben’ mit nachfolgender (im Frühnhd. entwickelter) trans. Verwendung?1
My own personal guess would be that it could be connected to the sharp, stinging sensation you experience when you don’t have access to enough oxygen to satisfy your system: when exercising, when out of breath, when diving (without aqualungs) and drowning, at high altitudes, and importantly also when undergoing asphyxiation. In German, this sensation is called Seitenstechen, an exact equivalent to English side stitch (‘stitch’ being also in English closely related to the noun ‘stick’—Scandinavian languages have equivalent expressions as well), so on some level there is a common Germanic connection between stinging/pricking/being sharp and the results of not being able to breathe.
So although English hasn’t developed a verb based on ‘stick’ that means ‘suffocate, etc.’, the most exact etymological cognate of the Dutch/German words is stick, and a purely formal cognate equivalent to Stickstoff is ‘stick stuff’. I wouldn’t recommend that you start referring to nitrogen as ‘stick stuff’, though.
1 I’m not the world’s best translator of German, but something like: “The semantic connection is unclear. Are we to assume that ersticken [suffocate] is from ‘killing through multiple stabs’ (cf. MHG si lāgen ersticket und verdorben ‘they lay stabbed and rotten/ruined’), with subsequent semantic shift to ‘die (from lack of air), kill (through deprivation of air)’ (as ¹DWB 10, 2, 2, 2742 assumes)? Or did ‘stick, plug into’ lead to the notion of ‘taking away someone’s breath through ‘plugging up’ their windpipe’ (cf. MHG ersticken und erworgen begunde er an dem beine), thus ‘getting your breath stuck and giving up the ghost’ with subsequent (and in Early Modern German extended) transitive usage?”