Do molecules like NH2 cation exist?
As @Mithoron said, $\ce{NH2+}$ ion is called parent nitrenium ion. The following text is an abstract from a paper1 which discussed its generation and various trapping pathways:

[...] (It) has been generated in various solutions through photolysis of 1-amino-2,4,6-triphenylpyridinium perchlorate. It was
found that the hydrocarbons, toluene and triphenylmethane trap
$\ce{NH2+}$ through two independent pathways. (1) free radical pathway
which leads to bibenzyl (in the case of toluene) or the trityl radical
(in the case of triphenylmethane). (2) hydride transfer
pathway which leads to benzyltoluene (in the case of toluene) or
triphenylmethanol (in the case of triphenylmethane). Results from dilution, triplet sensitization, and EPR experiments lead to the assignment of the hydride transfer pathway  to the singlet state of $\ce{NH2+}$ and the radical pathway to the triplet state of $\ce{NH2+}$ [...]
$$\ce{PhCH2+ + NH3 &lt;-[PhCH3] ^{1}NH2+ -&gt; ^{3}NH2+ -&gt;[PhCH3] NH3+^{.} + PhCH2. }$$

There is also another paper2 which discuss about the reaction of $\ce{NH+, NH2+}$, and $\ce{NH3+}$ ions with $\ce{H2}$ at low temperatures which leads to formation of ammonium ions (the general pathway of $\ce{NH4+}$ generation in interstellar environment):
$$\ce{N+ -&gt;[H2] NH+ -&gt;[H2] NH2+ -&gt;[H2] NH3+ -&gt;[H2] NH4+}$$
This reaction scheme is very much temperature dependent as it can lead to different pathways. For instance, reaction between $\ce{NH+}$ and $\ce{H2}$ has two pathways:
$$
\begin{align}
\ce{NH+ + H2 &amp;-&gt;[k^{a}_{NH^{+}}] NH2+ + H} &amp;\quad \Delta H^{\circ} = \pu{-1.9 eV}  &amp; \tag{1}\\
\ce{NH+ + H2 &amp;-&gt;[k^{b}_{NH^{+}}] H3+ + N} &amp;\quad \Delta H^{\circ} = \pu{-0.8 eV} &amp; \tag{2}\\
\end{align}
$$
The former pathway occurs at 300 K while the latter one occurs at 15 K.
Reference

On the Solution Chemistry of Parent Nitrenium Ion NH2+:  The Role of
the Singlet and Triplet States in Its Reactions with Water,
Methanol, and Hydrocarbons Sanjay Srivastava, Michael Kercher, and
Daniel E. Falvey The Journal of Organic Chemistry 1999 64 (16),
5853-5857 DOI: 10.1021/jo990324+
Reaction of NH+, NH2+, and NH3+ ions with H2 at low temperatures
The pathway to ammonia production in the interstellar medium
S. Rednyk, Š. Roučka, A. Kovalenko, T. D. Tran, P. Dohnal, R. Plašil and J. Glosík, A&amp;A
Volume 625, 2019, DOI: 10.1051/0004-6361/201834149