Do I need more than one random slope? When constructing a GLMM in R, do I need more than one random slope if I "see" that slopes differ for multiple continuous variables?
In my case, I am analysing the number of plant species (S) found across several large areal units (biomes) in southern Africa. I have several continuous predictors, mainly climatic parameters. For this I use a (G)LMM formula like this: 
myModel <- lmer(S ~ X1 + X2 + X3 + (1|biome), ...) # random intercept model

Now, I see that slopes for biomes differ strongly:  

Hence: I would adjust my formula to:
myModel <- lmer(S ~ X1 + X2 + X3 + (X1|biome), ...) # random intercept - slope model

However, if I now see that also X2 and X3 have differing slopes across the variables, should my model look like this?
myModel <- lmer(S ~ X1 + X2 + X3 + (X1|biome) + (X2|biome) + (X3|biome), ...) 
# random intercept - (multiple) slope model

I ask this question, because I am simply not sure whether this last formula makes the model too complicated or might be not "allowed" in a statistical sense, in short: I am simply "not sure" and I did not find any examples where more then one slope was estimated.
 A: It does make sense, but you have to be a little bit careful in setting up the model.  The way you've written the model,
 S ~ X1 + X2 + X3 + (X1|biome) + (X2|biome) + (X3|biome)

it implicitly incorporates an intercept term with each random effect slope.  You could write it as
 S ~ X1 + X2 + X3 + (1|biome) + (X1+0|biome) + (X2+0|biome) + (X3+0|biome)

which will estimate the intercept and all of the slopes separately. (It might be a good idea to center your covariates as recommended by Schielzeth 2010 ...)
Alternatively, in principle (but see below for caveats) you could use
 S ~ X1 + X2 + X3 + (X1+X2+X3|biome)

which would fit correlations among the slopes.
More fundamentally, however, I would consider (recommend?) fitting the among-biome variation as fixed effects rather than random effects, 
 S ~ (X1+X2+X3)*biome

(then you could just use lm instead of lmer).  Because you only have samples for 6 biomes, you will be estimating random-effects variances (and in the case of (X1+X2+X3|biome), a 4x4 random-effects variance-covariance matrix) from only 6 parameters.
One more comment: from your data, it looks like you have multiple observations with the same covariate (temperature range) value, which suggests that you are getting multiple observations from the same site.  I would think about incorporating site as a random effect ...
