Can one neuron spike inhibit and excite at the same time?
The textbook convention is that neurons release only one type of chemical from their synaptic terminals. This is known as Dale's Principle (https://en.wikipedia.org/wiki/Dale%27s_principle), and would exclude the possibility that neurons can release both inhibitory and excitatory neurotransmitters. 

Examples have been found of neurons which co-release two types of neurotransmitters, such as Glutamate and Dopamine [1] or GABA and Dopamine [2]. The effect of Dopamine depends on the types of Dopamine receptors in the target neuron, is generally modulatory, and can not be classified as purely excitatory or inhibitory [3], which is what your question asked. There is no evidence for neurons which release purely inhibitory and excitatory neurotransmitters, such as GABA and glutamate respectively. 

References:

[1] Sulzer, D., and S. Rayport. "Dale's principle and glutamate corelease from ventral midbrain dopamine neurons." Amino acids 19.1 (2000): 45-52.

[2] Tritsch, Nicolas X., Jun B. Ding, and Bernardo L. Sabatini. 
"Dopaminergic neurons inhibit striatal output through non-canonical release of GABA." Nature 490.7419 (2012): 262-266.

[3] Romanelli RJ, Williams JT, Neve KA (2009). "Chapter 6: Dopamine receptor signalling: intracellular pathways to behavior". In Neve KA. The Dopamine Receptors. Springer. pp. 137–174. ISBN 1-60327-333-6.