Do self-exciting neurons exist?
Short Answer

Yes, autapses exist, though the role of excitatory autapses in particular is unclear.

Long Answer

A lot of your assumptions are wrong for biological neurons (I'm suspecting you have a background in artificial neural networks but that might be inaccurate). These don't directly impact your question of whether these connections exist, but I think they are important for understanding how they could function, which is just as important if not more so.


The 'refractory period' refers to a period in which a cell cannot or is less likely to fire an action potential. Cells can receive inputs freely during their refractory period.
Neurons are not single compartments. Action potentials are generated near the soma, and although they can propagate into the dendrites, synapses out on the dendrites can be fairly electrically isolated from the soma, and synaptic potentials in dendrites can take some time, easily a ms or two, to affect the soma, and they also last for several ms. They can also affect the potency of other nearby synapses.
Although axons can be cms or more in length, dendrites are rarely found and further than 1 mm from a soma in the CNS (there can be exceptions, of course; I am mostly thinking of neocortex), and most dendrites are within 100-200 um from the soma. So travel distances are much shorter than your estimate. However, synaptic transmission itself is somewhat slow, so you can add another .5 ms, and your estimate for speed is quite fast for the brain; the link you gave applies to peripheral fibers and spinal cord to brain transmission only (note they are talking about "nerves"), so your overall estimate of 1 ms from spike to autapse turns out to be reasonable, but for different reasons.
It isn't meaningful to talk about refractory periods for most synapses. Synapses can display short-term depression and/or facilitation but that isn't the same as a refractory period. Only at particular synapses containing a single release site does it make any sense to talk about refractory periods.
Cells are not activated by an individual synapse, with a few exceptions in very specialized brain areas. So just because a cell has been made to fire tells you nothing about whether a particular synapse was activated. In fact, unless that cell has fired previously very recently, you can be confident that an autapse would not have contributed to that particular firing event.


Evidence for excitatory autapses

You specifically asked about self-excitation, but it's important to know that inhibitory self-synapses also exist, and they are much more common between GABAergic inhibitory cells than between excitatory cells (Bekkers, 1998). Inhibitory autapses make a lot of functional sense because they serve as direct negative feedback.

However, excitatory autapses do exist and they may have function significance(Bekkers, 2009), rather than just being rare "accidents" that do little. In Aplysia, for example, there are autapses that cause an excitatory plateau potential during feeding behavior, maintaining activity for a long duration (Saada et al., 2009).

In mammalian neocortex, excitatory autapses have been observed but their function is unclear. Their properties can be a bit different from other synapses (Liu et al., 2013). Spike-timing dependent plasticity would suggest that autapses should shrink and go away, but they don't, which suggests they might have some functional utility, though it isn't understood yet. There are many computational attempts to find some function, which I won't list here but you can easily find by searching a reference like Google Scholar for recent articles about "autapses."

Experimentally, the role of autapses in vivo is difficult to study because there is no specific way to selectively suppress autapses outside of a computational environment, or a reduced prep where individual cells can be activated without activity in the rest of the network.



References

Bekkers, J. M. (1998). Neurophysiology: Are autapses prodigal synapses?. Current biology, 8(2), R52-R55.

Bekkers, J. M. (2009). Synaptic transmission: excitatory autapses find a function?. Current Biology, 19(7), R296-R298.

Saada, R., Miller, N., Hurwitz, I., &amp; Susswein, A. J. (2009). Autaptic excitation elicits persistent activity and a plateau potential in a neuron of known behavioral function. Current Biology, 19(6), 479-484.

Liu, H., Chapman, E. R., &amp; Dean, C. (2013). “Self” versus “non-self” connectivity dictates properties of synaptic transmission and plasticity. PloS one, 8(4), e62414.