The source of your confusion is that a living cell is not the most basic organism, it is in fact very complex. The most basic organisms running around now are viruses, or arguably prions, and they are very much simpler than cells. They are much more like self-replicating molecules. The one hundred base pair creature would be a very simple virus-like organism.
I guess that solves the evolution of the virus or prion, but what about a cell?
It does not have to be consecutive. It could happen that there are one million primitive self-replicating molecules, and one hundred thousand mutate badly and die out, and only one develops a positive mutation.
Since that one with the positive mutation has a competitive advantage, its descendants will tend to outnumber or even wipe out the ones without the advantageous mutation. So then we have one million mutants, and maybe one of those mutants develops the next advantageous mutation and so on and on. The typewriting monkeys don't press certain keys more than others, but it might look like they do because we throw out all the pages where they hit the wrong keys.
You have to think of it more practically than that. Even if the monkey's avoided bashing the key board with their palms or several fingers at once and were trained to only press a single key at a time (thereby reducing the chances of errors) it would be impossible for a monkey to create on of Shakespears fat books, in fact, they wouldn't even come close. Think of how many billions of key strokes it would take to create something even remotely resembling a paragraph, riddled with spelling errors, but close enough to be legible. Now let's say that by some miracle, one monkey manages to make it to his third or fourth, perfectly written paragraph... then he makes a couple errors and all his work is whiped out.
There are two main problems here that aren't likely to be overcome without some sort of a discriminating mechanism between the 'good' and 'bad' mutations.
First, a 'good' mutation may be good in the context of hundreds of thousands of other
good mutations, but on it's own it can't provide any advantage in the early stage of developement of DNA. An example of this would be an eagle who's DNA is altered by solar debris let's say, it get's sliced at some point and reconnects. Let's say this section of the DNA controls the size of the eagle's talons, which could very likely give the eagle an advantage at this point, but don't forget, there are likely hundreds of thousands of instructions that make the talon itself
already in place.
Before DNA made up any useful aspects of an organism it required hundreds of thousands of positive mutations to get there. A few positive mutations, or even hundreds wouldn't have made enough of a change to do anything at all at this point because there was no 'functioning' sections of the strand that could be modified yet.
Second problem is this, it only takes a few negative mutations to destroy the whole thing apparently and chances are the negative are 50/50 with the positive. This in essence would disable the positive mutations in an 'original blend' of DNA to reach any useful number before it was basically destroyed due to a number of negative mutations. To better illustrate what I mean, imagine you had a coin and a chart of about 500,000 different states listed, each being either heads or tails. If you were to flip a coin enough times, discarding the negative flips, you would eventually match all the states or 'choices' written on the chart, but what if I introduced a new rule, make 10 or more errors and you have to start all over again.
Mathematically speaking yes, it's "possible" for you to flip a coin 500,000 times and match each random state depicted on the chart perfectly, but realistically speaking it's more than a long shot, it's basically impossible because of the rule that nearly all your flips must be
consecutively correct in order to avoid making any errors. Sure, you might produce 50-60 or maybe even by some miracle, over 200-300 consecutive flips after several billion years that are truly
random and matching each state on the chart, but you make a few errors and you'll have to start all over!!
Now let's say there are trillions of you, flipping for trillions of years and one of you make it to 500 flips (to give you an idea how unlikely this is, it's basically the same thing as flipping heads or tails 500 times in a row)... but wait, we are talking about
half a million flips needing to be right and the chances of breaking the code long before you even get
close to making it are simply too high. There would have to be some discriminating mechanism that improves your chances of flipping the right state and lessening your chances of flipping the wrong state if you are to reach that high of a number.