We are looking at the steady state, so transitional dynamics don't matter. As the economy develops and approaches the steady state many forces act upon the savings rate. On one hand, agents will save less when they are poor due to the form of the utility function (declining marginal utility, consumption smoothing), on the other hand, the return on saving is higher when the capital stock is low - which actually should lead to a higher savings rate in poor economies. How the saving rate (and therefore output growth) develops over time for different countries depends on quite a lot of paramters.

However, the savings rate will be constant in the long-run (but not necessarily the same for different countries) and so will be output growth.

Maybe you have heard of the Solow model (you will find lots of information on the internet) where the savings rate is exogeneously given. Here it makes more sense to ask what happens if the savings rate is incrased by x%. The answer is that output growth would temporarily increase but after a while output growth would be the same as before. So an incrase in the savings rate only has a level effect. 

Since there is no uncertainty in this model, I'm not sure that it adds anything to interpret theta as the coefficient of relative risk aversion. Its sole effect is that of (the reciprocal of) the elasticity of intertemporal substitution. 

Lucas notes in a footnote* that "the coefficient of risk aversion is sometimes called the intertemporal elasticity of substitution. Since all the models considered in this chapter are deterministic, this latter terminology might be more suitable."

When thinking about the curvature of the utility function (u(c) → ln(c) as θ → 1 and u(c) → c -1 as θ → 0) I immediately categorize in risk averse, risk neutral, and risk loving. Maybe that's why I also tend to denote θ the coefficient of (relative) risk aversion. I do not spend more time thinking about the form of the utility function. Btw, it has never happend to me that I met a student who could correctly explain the Elasticity of Substitution.

*after denoting θ (actually σ - he uses a different notation) the coefficient of (relative) risk aversion