The RIP-nomenclature is used to distinguish between different runs of the same scenario within a modeling center rather than to indicate any similarity across modeling centers. Strictly it's within and across models not centers, because some centers have more than one model, e.g., the MIROC center submitted models MIROC5, MIROC4h, MIROC-ESM, etc. So r1i1p1 for some scenario is typically just the ensemble member that happened to be run first for each model. One caveat is that whichever historial ensemble member you pick for each model, you should use the same member from the corresponding RCP scenario, e.g., historical r3i1p1 should join seamlessly with rcp8.5 r3i1p1:
A recommendation for CMIP5 is that each so-called RCP (future scenario) simulation should when possible be assigned the same realization integer as the historical run from which it was initiated.
From Taylor et al (2012) CMIP5 Data Reference Syntax (DRS) and Controlled Vocabularies (pdf)
There's loads more detail in that Data Reference Syntax doc that I've linked to, and a summary of it on the IPSL website. Karl Taylor gave a presentation that summarised it even more succinctly:
- r = “realization”: simulations started from equally likely initial conditions
that lead to equally likely realizations of the true climate trajectory
- i = “initialization”: only used in decadal predictions, to distinguish among
different initialization procedures
- p = “physics”: to identify simulations that are very closely related (e.g.,
“perturbed physics” ensemble members or simulations forced by slightly
In practice, the "initialization" and "physics" ensemble members are a bit specialist and the vast majority of CMIP5 users just stick to the rNi1p1 members, and most of them probably just stick to r1i1p1 from each model (e.g, Peng et al, 2019) for simplicity. You should feel free to use any of the "realisation" members from each model, and you could pick r1i1p1 from one model and r10i1p1 from another. I'd advise sticking to one ensemble member from each model though, to avoid your analysis over-representing the forced responses of models that submitted more realizations to the CMIP5 database.
To address your final question, it's more common to look at the variance across the models for a given scenario. But there are also plenty of studies that use the variance within each model ensemble to estimate the uncertainty of a model property due to internal variability, e.g., emergent constraints on climate sensitivity. It really depends what you're investigating.