Understand what reference category in Linear Mixed Model mean for interpretation Since I am not very familiar with LMM, I am following this  to gain some understanding.
Assume I have very simple model

Y ~ 1 + A + (1 + A|B)

Where B has 10 classes.
When I run this model, lmer automatically assumed group 0 as baseline according to document.
summary(model)

Tells me the fixed effect part which are the Intercept  & Coefficient of A.
coef(model)

Tell me the random part which  are intercept and coefficient of A for each of 10 classes in B
Q1. Is my understanding that fixed effect is just mean of all the random effect? OR, are they referring to intercept & coefficient for class 0?
Q2. If for Q1 foremost understanding is correct, should not the random effect for class 0 be 0 since it is the reference category?

Let's take a look at some output from the same model on a toy dataset:
Linear mixed model fit by REML ['lmerMod']
Formula: Y ~ 1 + A + (1 + A | B)

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 B        (Intercept) 9.3832   3.0632        
          A1          4.6398   2.1540   -0.82
 Residual             0.4091   0.6396        
Number of obs: 40, groups:  B, 10

Fixed effects:
            Estimate Std. Error t value
(Intercept)   9.9420     0.9792  10.153
A1            2.2483     0.7106   3.164


When I run this model, lmer automatically assumed group 0 as baseline according to document.

Yes, this has nothing to do with lmer itself. This is the standard treatment of a categorical variable in R. The estimate for the reference level is contained in the estimate of the intercept.

Q1. Is my understanding that fixed effect is just mean of all the random effect? OR, are they referring to intercept & coefficient for class 0?

No, the mean of the random effects is zero. The random effects are offsets from the fixed effects, but the overall "effect" is zero.
Here, the fixed intercept is the expected value of the response in group 0, and the fixed effect for A is the difference in the expected value of the response between group 0 and group 1
