Are there natural-forcings climate projections over the 21st century?
You’re right that there were no future natural forcings experiments defined in the CMIP5 protocol.  The reason being simply that the modeling groups thought that other experiments were more deserving of their limited computing time.  Compare this with the much stronger call for the historicalNat and historicalGHG experiments, which were required for detection and attribution studies.  Individual modeling centers may have run future natural forcings experiments, but that will be hit and miss, and there wasn’t an experiment name to lodge the outputs to in the CMIP5 database.  Incidentally, the situation with the RCP (all forcings) experiments themselves was actually a bit imperfect, because the implementations of future natural forcings were left to the discretion of each modeling group:


  
  … It is recommended that either volcanic aerosols should be omitted entirely from the control and future runs, or, alternatively, the same background aerosol should be prescribed in both runs.
  It is recommended that some representation of the solar cycle be included in the 20th and 21st century simulations, though that is left up to the discretion of the modeling groups.
  
  
  Source: Taylor et al (2011).  A summary of the CMIP5 experiment design. (pdf)


There was a further recommendation for how future solar irradiance forcing should be represented: 


  In CMIP5, climate projections were based on a stationary-Sun scenario, obtained by simply repeating solar cycle 23, which ran from April 1996 to June 2008 (Lean and Rind, 2009).
  
  Source: Matthes et al (2017).  Solar forcing for CMIP6 (v3.2), Geosci. Model Dev., 10, 2247-2302, https://doi.org/10.5194/gmd-10-2247-2017


But, in practice, groups did slightly different things from each other, so you usually have to check the output file metadata and any associated journal papers for individual model runs to get an idea of the details.

This is clearly not a crazy question though, because in CMIP6 some future natural forcings experiments have been defined, albeit in the lower priority tiers. Specifically, the Detection and Attribution MIP (DAMIP) includes ssp245-GHG and ssp245-nat experiments, and the volcano MIP (VolMIP) has the volc-cluster-21C experiment.  If you dig down into those you’ll find that future solar variability is derived more or less the way you describe, i.e., projecting forward based on a set of statistical time series models trained on past variability.  Note, however, that Matthes et al are careful to point out that this is a projection not a prediction:


  As of today, predicting solar activity up to 2300 is very challenging, if not impossible… The unusually long solar cycle number 23 that ended in 2009, and the weak one (no. 24) that followed came as a surprise, and manifested our evident lack of understanding of the solar cycle.


Conversely, future volcanic forcing in ssp245-nat will be prescribed as a near-constant background forcing:


  Volcanic forcing will be ramped up from the value at the end of the historical simulation period (2015) over 10 years to the same constant value prescribed for the piControl simulations in the DECK, and then will be kept fixed.
  
  Source: O'Neill et al (2016). The Scenario Model Intercomparison Project (ScenarioMIP) for CMIP6. Geosci. Model Dev., 9, 3461-3482, http://doi.org/10.5194/gmd-9-3461-2016


So why not a statistical model for volcanic emissions?  This forcing is episodic rather than cyclic and is prescribed to the CMIP models as several variables on latitude-height-time grids.  As you hint at in your question, this is too many degrees of freedom to pin down satisfactorily from just the 1850-2005 record.  Even if it could be done, you then have the problem of how to use the statistical model, given that each climate model will only run a few realisations of a particular projection.  Do you generate one scenario to be shared by all models, or do you let each group generate its own scenarios using the statistical model?  Either way, you can’t explore enough of the parameter space to avoid ending up with a CMIP6 ensemble containing decadal variability that is strongly influenced by forcing scenarios that are extremely unlikely to happen.

As I understand it, the current assumption is that volcanic forcing is stationary and, in the absence of knowing the future sequence of eruptions, its effect on slowly varying climate components like ocean heat content is best represented by a constant mean forcing.