Can one neuron spike inhibit and excite at the same time? Is it possible to have a neuron that in some synapses releases inhibitory neurotransmitters and excitatory in others (everything triggered by the same spike) ?
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.