How do i use R to input a number of grouping values and recieve an output of observations in a 8x6 grid?

Here's an approach with the help of dplyr, grid and gridExtra.

Please excuse the hot mess that is my variable naming convention.

Your data was not complex enough to make a good system, so I generated some random data. Find that at the very end.

First, lets define our litters and filter the mouse data.

library(dplyr)
library(grid)
library(gridExtra)

geno.litters <- c("B23701-2", "B23744-1", "B23844-1","B23944-1")
mice <- data %>% 
            filter(Litter_ID %in% geno.litters) %>%
            arrange(Litter_ID,MouseID) %>%
            split(.,.$PCR_Temp)

mice is now a list of the mice split into plates by PCR temperature.

Let's define a custom function to add positive and negative controls and duplicate rows for those genotypes that need duplicates. We can apply that function to every list element with lapply.

addControlSlots <- function(x){
  genotypes <- unique(x$Genotype)
  genotype.dfs <- list()
  for ( i in seq_along(genotypes)){
    litter.mice <- x[x$Genotype == genotypes[i],]
    litter <- litter.mice[1,"Litter_ID"]
    Temp <- litter.mice[1,"PCR_Temp"]
    litter.mice <- rbind(litter.mice,litter.mice[litter.mice$Rxns == 2,])
    litter.mice <- litter.mice[order(litter.mice$MouseID),]
    control.rows <- data.frame(Litter_ID = litter, MouseID = c("PosCont","NegCont"),Gender = NA,Genotype = genotypes[i], PCR_Temp = Temp, Rxns = 1)
    genotype.dfs[[i]] <- rbind(litter.mice,control.rows)
  }
do.call(rbind,genotype.dfs)
}

processed.temps <- lapply(mice,addControlSlots)
processed.temps[[2]]
#$Z
#    Litter_ID MouseID Gender  Genotype PCR_Temp Rxns
#1    B23701-2   ZO960      F   zDC.Cre        Z    1
#2    B23701-2   ZP810      F   zDC.Cre        Z    1
#3    B23701-2   ZP992      M   zDC.Cre        Z    1
#4    B23701-2 PosCont   <NA>   zDC.Cre        Z    1
#5    B23701-2 NegCont   <NA>   zDC.Cre        Z    1
#...15 more rows

We now have controls after every genotype.

Now let's define a function to fill in the PCR plate. And again apply it to the list.

makePCRPlate <- function(x){
  mouse.vector <- as.character(x$MouseID)
  plate.vector <- rep(NA,6*8)
  plate.vector[1:length(mouse.vector)] <- mouse.vector
  wide <- matrix(plate.vector,nrow=2,byrow = FALSE)
  rbind(wide[,1:8],wide[,9:16],wide[,17:24])
} 

pcr.plates <- lapply(processed.temps,makePCRPlate)
pcr.plates[[2]]
#     [,1]      [,2]      [,3]      [,4]    [,5]    [,6]    [,7]    [,8]   
#[1,] "ZO960"   "ZP992"   "NegCont" "ZO214" "ZP333" "ZP455" "ZP478" "ZQ130"
#[2,] "ZP810"   "PosCont" "ZO214"   "ZP333" "ZP455" "ZP478" "ZQ130" "ZQ875"
#[3,] "ZQ875"   "NegCont" NA        NA      NA      NA      NA      NA     
#[4,] "PosCont" NA        NA        NA      NA      NA      NA      NA     
#[5,] NA        NA        NA        NA      NA      NA      NA      NA     
#[6,] NA        NA        NA        NA      NA      NA      NA      NA    

We can see that the samples have been filled in in the zig-zag pattern.

Now let's use grid to make a .pdf file with the layouts.

pdf("MyPCRPlates.pdf")
for(i in seq_along(pcr.plates)){
  grid.newpage()
  grid.table(pcr.plates[[i]])
  grid.text(paste0("PCR Temp ",names(pcr.plates)[i]),y = unit(0.9,"npc"))
}
dev.off()  

The .pdf file should have a page for each temperature.

Data

set.seed(1)
data1 <- data.frame("MouseID" = paste0("Z",sample(c("O","P","Q"),size = 50,replace = TRUE),round(runif(50,1,999))),
Litter_ID = sample(c("B23701-2", "B23744-1", "B23744-2", "B23745-2", "B23844-1","B23944-1", "B23944-2", "B23951-1"),size=50, replace = TRUE),
Gender = sample(c("F","M"), size = 50, replace = TRUE))

data2 <- data.frame(Genotype = c("zDC.Cre","Villin.Cre","Villin.Cre","zDC.Cre","K14.CreER","Rosa.TSLP","Rosa.TSLP","K14.CreER"), 
                    Litter_ID = c("B23701-2", "B23744-1", "B23744-2", "B23745-2", "B23844-1","B23944-1", "B23944-2", "B23951-1"),
                    PCR_Temp = c("Z","Y","Y","Z","Y","Z","Z","Y"),
                    Rxns = c(1,1,1,1,1,2,2,1))
data <- merge(data1,data2)
data
#   Litter_ID MouseID Gender   Genotype PCR_Temp Rxns
#1   B23701-2   ZP810      F    zDC.Cre        Z    1
#2   B23701-2   ZP992      M    zDC.Cre        Z    1
#3   B23701-2   ZO960      F    zDC.Cre        Z    1
#4   B23744-1   ZO122      F Villin.Cre        Y    1
#5   B23744-1   ZQ259      F Villin.Cre        Y    1
#... 45 more rows