Nested random effects in lme4 R Background:
I have data on time to infection across multiple sites across a gradient. The design involves 2 latitudes (In and Out) with sites 1 and 2 nested within “In” and sites 3 and 4 nested within “Out.” Within each of the four sites I have three transects and within each of the transects I have 5 plates.
Each of the plates is explicitly nested within each of the transects and each of the transects is explicitly nested within each of the sites. 
I am interested in knowing if there is significant difference in infection across the In and Out gradient as well as across the four sites. Therefore I have made both gradient and site fixed factors (also because they have fewer than 4 levels). I am also interested in looking to see how much of the total variance is due to the transects and the plates, although this is secondary to the first question.

Potential model specifications: 
a <- glmer(Response~Gradient+Site +(1|Site), 
           data=surv, family="poisson", nAGQ=7)) 
b <- glmer(Response~Gradient+Site +(1|Transect)+(1|Plate), 
           data=surv, family="poisson") 
c <- glmer(Response~Gradient+Site +(1|Transect), 
           data=surv, family="poisson", nAGQ=7))
d <- glmer(Response~Gradient+Site +(1|Transect/Plate), 
           data=surv, family="poisson", nAGQ=7))
e <- glmer(Response~Gradient+Site +(1|Transect/Plate)+(1|Gradient/Site), 
           data=surv, family="poisson"))

Questions:  


*

*I have seen examples were the lowest level of nesting was not put into the model specification. For example, not putting in “plate” in my case (model a and c). How do you know to include the lowest level of your sampling design?  Would it be correct to compare the AICc values for each model and drop the models that include the lowest level of nesting (plate) if they have AICc values larger than the models that exclude them? My understanding is that comparing AICc is valid here as the fixed effects are the same across the models and only the random effects are changing.

*If the answer to question 1 is yes, it is justifiable to discard models b, d, and e as they have higher AICc values than a, c. Now to choose between a and c. Including “site” as both a fixed and random factor significantly changes the model outcomes compared to the other models. What would be the interpretation of having both site as a random and fixed effect? I have seen other models specified this way, that is why I am asking.

*Is it justifiable to break up the models into two?  I ask because I often get the message: " fixed-effect model matrix is rank deficient so dropping 1 column / coefficient” which makes me think I don’t have enough data for the number of terms I have included in the model. 
For example: 
f <- glmer(Response~Gradient+(1|Site),  data=surv, family="poisson", nAGQ=7)) 
g <- glmer(Response~Site+( 1|Transect), data=surv, family="poisson")) 

Note: I will check for overdispersion as I am using a Poisson distribution.
Example data:
Gradient    Site    Transect    Plate   Response
In  Site1   Transect1   6   11
In  Site1   Transect1   18  27
In  Site1   Transect1   28  51
In  Site1   Transect2   3   19
In  Site1   Transect2   29  19
In  Site1   Transect2   36  27
In  Site1   Transect2   43  51
In  Site1   Transect3   19  19
In  Site1   Transect3   25  27
In  Site1   Transect3   9   19
In  Site1   Transect3   46  NA
In  Site1   Transect3   49  19
In  Site2   Transect4   5   27
In  Site2   Transect4   16  20
In  Site2   Transect4   24  20
In  Site2   Transect4   42  27
In  Site2   Trasect5    20  20
In  Site2   Trasect5    33  20
In  Site2   Trasect5    7   27
In  Site2   Transect6   2   20
In  Site2   Transect6   38  20
In  Site2   Transect6   48  27
In  Site2   Transect6   17  62
In  Site2   Transect6   21  20
Out Site3   Trasect7    4   19
Out Site3   Trasect7    26  19
Out Site3   Trasect7    27  51

 A: Let me try to answer Q3 first:
Is it justifiable to break up the models into two? I don't think so. Instead of finding a way around the message fixed-effect model matrix is rank deficient so dropping 1 column / coefficient, we should try to figure out why that happened. Usually, what this means is that you have not enough information to estimate the specified model.
I created an example that matches your experimental design above:
set.seed(23)
## YOUR DESIGN AS DEPITCTED IN THE IMAGE
surv <- data.frame(
  Gradient = rep(c("In", "Out"), each = 30),
  Site = rep(paste("Site", c(1:4), sep = ""), each = 15),
  Transect = rep(paste("T", c(1:12), sep = ""), each = 5),
  Plate = factor(sample(c(1:60))),
  # GET SOME RANDOM NUMBERS FROM POISSON DISTRIBUTION
  Response = rpois(60, 6)
)

Now using your model a for example, we can reproduce the message fixed-effect model matrix is rank deficient so dropping 1 column / coefficient (actually that's also the case for your models b, c, d, and e):
library(lme4)
model_a <- glmer(Response ~ Gradient + Site + (1|Site),
                 data = surv, family = "poisson", nAGQ = 7)

When you look at the fixed effects output (summary(model_a), also check: model.matrix(model_a)), you can see that Site4 was dropped. This has most likely to do with the fact that your fixed effects (factors and levels) cannot be represented as unique combinations of each other. Or in other words, you have levels of Site which are not represented in levels of Gradient. This is nothing new since you already specified that Site is nested within Gradient. Now, this also means that you cannot test whether levels of Site are significantly different. Instead, you would want to quantify and account for the (random-)effect of Site (and Transect) on the fixed-effect Gradient.
Here's how I would specify the model:
model_b <- glmer(Response ~ Gradient + (1|Site) + (1|Transect), 
                 data=surv, family = "poisson")
summary(model_b)

[showing only relevant output]

Random effects:
 Groups   Name        Variance Std.Dev.
 Transect (Intercept) 0        0       
 Site     (Intercept) 0        0       
Number of obs: 60, groups:  Transect, 12; Site, 4

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  1.81916    0.07352  24.743   <2e-16 ***
GradientOut -0.04987    0.10530  -0.474    0.636    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

When you look at the part of the output where is says Number of obs: 60, groups:  Transect, 12; Site, 4, you can see that the grouping matches your design. Also since your nesting is explicit, you don't need to specify nesting in your random-effects statement (using the \ sign) since it's implied by your data. The variance estimates of the random effects will probably change when you enter your real data.
As for Q1, I think that Plate itself isn't a level since it has no replicates and is also not repeatedly measured over time. The interaction of Transect:Plate would be the lowest level in your case, which is represented by (1|Transect) due to the explicit nesting structure in your data. 
As for Q2, since Site seems to be of interest to you (included as fixed effect in your models), it could be treated as such, if you reduce its levels from four to two and replicate it in the Gradient levels In and Out. However, I don't know whether this is possible because I don't know enough about your Site factor. If that's reasonable, you could do this:
surv2 <- data.frame(
  Gradient = rep(c("In","Out"), each = 30),
  Site = rep(paste("Site", c(1:2), 2, sep = ""), each = 15),
  Transect = rep(paste("T", c(1:12), sep = ""), each = 5),
  Plate = factor(sample(c(1:60))),
  # GET SOME RANDOM NUMBERS FROM POISSON DISTRIBUTION
  Response = rpois(60, 6)
)

model_c <- glmer(Response ~ Gradient + Site  + (1|Transect), 
                 data = surv2, family = "poisson")
summary(model_c)

and potentially also allowing for random slopes between levels of Site:
model_d <- glmer(Response ~ Gradient + Site  + (Site|Transect), 
                 data = surv2, family = "poisson")
summary(model_d)

Note: This won't give you the fixed-effect model matrix is rank deficient so dropping 1 column / coefficient message any more.
I don't think that you need to choose between models by comparing the AICs in this case here. I think the most useful model given the data is model_b and perhaps model_c or model_d if this is feasible. But maybe someone else can comment if that's incorrect.
