I am not sure what is missing for the log likelihood function as the log likelihood function is simply taking the log of the likelihood function by definition. There is not really anything to explain. So that we have ln(theta^n * e^(-theta*(sum of Yi))) = n*ln(theta) – theta*sum of Yi.

For the maxium estimator, we need to find the likelihood function maxium under what theta value which is the same as the loglikelihood function maxium, as log is an increasing function. Thus we take the derivative of the loglikelihood function and calculate under what value of theta will the derivative become 0 and it is reciprocal of the mean of Yi.

Last for the R code.

nloglik here is simply the result of the loglikelihood function.

function(x,theta) means the function has variable x and theta.

sum(-dexp(Y=Y,rate=theta,log=T)) here is to define the loglikelihood function.

optimize(f = nloglik ,Y = Y,interval=c(0,n)) is asking the computer to find to maximizer of nloglik, which is the loglikelihood function, Y = Y is to tell the computer input Y is fixed, interval=c(0,n) is to tell the computer which part of Y are we considering.

https://en.wikipedia.org/wiki/Maximum_likelihood_estimation

Here the E step is trying to find the theta maximizer for loglikihood function. Here Q(theta|theta (t)) = maxium theta (log(L(theta;X,Z))), in which X is the meassurements and Z is other coefficients for the distribution, which is not needed as exponential distributions only have one variable theta. 

I believe the theta here would be pretty small so that the meassurements would spread more evenly among the positive numbers instead of concentrating between 0 and 1. Or it would be unaccurate to only consider the integer part of the Ts. If you are still worried about this, we can replace Yi by Yi+0.5, but this would not our answer at all, as we are multiplying our loglikelihood function by a constant.

It depends on you whether to write that down or not. At last on my laptop, the code would print this out to tell us how the computer calculate theta. This comes from part a that we have proven that reciprocal of the mean of Y is the MLE for theta.