Has the number of natural disasters per year increased significantly in the years since human-caused climate change started?
We have no evidence to suggest that the rate or magnitude of earthquakes has, in general, increased: while earthquakes are having a greater effect on humans, this is entirely - as best we can tell - due to increased population density, and the propensity of people to build cities immediately above major fault planes (because they channel water to the surface along impermeable fault debris). Populations settling above the surface expressions of major faults isn't much of an issue when you're living in tents in a sparse community, but once you get high-density populations living in skyscrapers with poorly-enforced building regulations death and injury rates rise very rapidly. For more detail and onward references see the USGS FAQ on this topic.

For statistics on extreme weather phenomena/natural disasters, particularly relating to precipitation and flooding, the IPCC is an excellent resource. There is of course some degree of difficulty with reporting bias (as with earthquakes), but the IPCC are extremely rigorous in calculating - and providing - bounds on probabilities and certainties. In particular, see Working Group 2: Impacts, Adaptation, and Vulnerability; their summary for policy-makers has to say:


  Impacts from recent climate-related extremes, such as heat waves, droughts, floods, cyclones, and wildfires, reveal significant vulnerability and exposure of some ecosystems and many human systems to current climate variability (very high confidence).


For more details, see the final draft of WG2's contribution to the Fifth Assessment Report (AR5). Particularly worthy of highlight are the FAQ and chapter 18.4.3, which has to say:


  The last several decades have seen changes in the frequency and intensity of extreme weather events including extreme temperature, droughts, heavy rainfall, and tropical and extratropical cyclones with low to very high confidence, depending on the type of extreme event (IPCC, 2012, WGI AR5 Chapter 2). However, the impacts of extreme weather events also depend on the vulnerability and exposure of systems. It is possible that climate change can affect vulnerability and exposure, but typically both are primarily influenced by non-climate confounders, most notably economic development.