Answering the question in the manner that you are asking for would require quite an exhaustive list. However, a fundamental concept in all of this is having a "leak" channel.
NALCN is the only nonselective channel found in the 24-TM channel family and is equally permeable to Na+, K+, and Cs+. 
The majority of the ions transported by the channel are sodium. This slow influx will have the same effect on voltage over a longer period of time as would a rapid influx, a depolarization of the membrane.
NALCN is found in all animals studied, from humans to D. melanogaster, C. elegans, snails, sea urchins, and the placozoan Trichoplax adhaerens.
So, it is likely that this channel is conserved in the evolutionary ladder.
NALCN mutant mice have a severely disrupted respiratory rhythm and die within 24 hours of birth. Brain stem-spinal cord recordings reveal reduced neuronal firing. The TTX- and Cs(+)-resistant background Na(+) leak current is absent in the mutant hippocampal neurons. 
This addresses your other question, this leakage makes the membrane voltage more permissive to initiation of rhythmic discharge and other oscillatory behaviors.
As to whether anything fires "randomly", I would say that neurons can fire with a given distribution , but this is governed by all the different channels which operate and their conductances (inverse of resistance) rather than just by chance alone. Remember, if a neuron fires randomly, it's not likely to have much effect on a post-synaptic cell unless it's time-locked to a series of other action potentials arriving at the same spot.
 Ren, D. (2011). Sodium Leak Channels in Neuronal Excitability and Rhythmic Behaviors. Neuron, 72(6):899-911. http://www.ncbi.nlm.nih.gov/pubmed/22196327
 Lu B, Su Y, et al (2007). The neuronal channel NALCN contributes resting sodium permeability and is required for normal respiratory rhythm. Cell, 129(2):371-83. http://www.ncbi.nlm.nih.gov/pubmed/17448995
 Webb TJ, Rolls ET, Deco G, Feng J. (2011) Noise in attractor networks in the brain produced by graded firing rate representations. PLoS One, 6(9):e23630. Free PDF