To answer your question more precisely there would have to be more details, but I understand that it would probably be too invasive privacy-wise.

So let me try to approach this from a general POV.

I have previously written about the children (incl. Jewish children) in Auschwitz here and I will be using this in the answer.

1. There are several groups of very small Jewish children who were not killed in Auschwitz and thus consitute an exception to the rule. You can see many of them in the liberation footage and photos.

Specifically, such children usually came with the transports that did not undergo selections.

An example of one such transport would be the so-called Phillips transport:

The train arrived in Auschwitz on June 6, the day of the allied invasion in Normandy. The acting commandant of Auschwitz Rudolf Höss was informed that a transport of trained radio technicians would be arriving in Auschwitz. The message came from Gerhard Maurer, head of the department for labour deployment at the SS Economic Administrative Main Office who had been contacted by Ewald Löser, a senior Philips executive in Berlin. Thus the Jews from Vught did not undergo a selection and were kept alive.

A group of Jewish transports from labor camps in the Generalgouvernement at the end of July, beginning of August of 1944 underwent no selection upon arrival. This includes the transport from Starachowice on 30.07.1944, the transports from Pionki and Blizyn on 31.07.1944, the transport from Kielce on 02.08.1944 and the transport from Ostrowiec on 03.08.1944. The most probable reason is that these GG transfers were seen as "internal" transfers from labor camps on an otherwise "judenrein" territory: it was assumed that most Jews unfit for work would have already been eliminated in the previous labor camps by the time of the arrival in Auschwitz, and it would be a waste of manpower to stage a selection for such transports for a few people who might be unfit for work.

Many of the children you see in the Soviet footage are from these exceptional transports.

Yet another group were the twins selected by Mengele for his "research" and experiments.

2. The only group of Jews that did not get tattoed in Auschwitz as a rule were the so-called transit Jews or Durchgangsjuden. Most of them were from Hungary, though some arrived from elsewhere, like Viktor Frankl.

They stayed in the camp for a while, usually in the "depot" section called Mexico, without becoming the camp inmates (hence no camp numbers and no tattoos), before being sent to various work camps (though they also could get selected for gassings).

Usually this wouldn't apply to small children, who were unable to work, though I suppose there always could be an exception for some reason (and that's where details would be useful).

3. Typical Auschwitz latrines looked like this: https://upload.wikimedia.org/wikipedia/commons/0/06/Auschwitz-Birkenau_latrine.jpg

I suppose a small child could have a fear of falling in there.

For a time open ditches were used (e. g. during the building of the camp), but I don't think this applies here.

4. No, not all prisoners were sent on death marches, the ill ones or small children, basically anyone unable to march, were supposed to stay in the camp, though in fact many inmates who were not supposed to march joined the marches anyway because they feared that those who stayed behind would be murdered (see e. g. A. Strzelecki, Endphase des KL Auschwitz, 1995, pp. 144, 248).

Apparently, the Nazis did want to murder most or all of the remaining inmates but in the last moment did not do it for whatever reasons (like the risk of being encircled by the Red Army; ibid., pp. 249-251).

That's why you can see the survivors in the Soviet liberation footage.

You could always try to reach out to the Auschwitz Museum, in my experience they are very helpful.

I can only add something to Nr 3 and I hope it's not a too-small snippet that gets removed before you read it, but other camps such as Bergen-Belsen would use vats as latrines during typhus epidemics when basically everyone had heavy diarrhea, so it's common practice

Most of your questions are going to depend on where she lived, both what country and if it was rural or urban. Also, if she was moved to a ghetto first, or if she was immediately moved into Auschwitz.

Some kids weren't immediately killed because of the shear number of prisoners being processed and killed. There was just a post I read on Reddit yesterday about a man who escaped as a four year old along with about 50 other children because they hid when the camp guards rounded up all the children to kill them.

Not everyone was sent on a death march. Some were transported in boxcars and trucks from their home to the ghetto and the ghetto to the death camp. Some were sent to a different work camp and then to a death camp. Finding out the specific towns, camps, and ghettos, and the dates of your grandmother's arrival and departure will help you put together the story of her survival.

I don't know why she wasn't tattooed. It's possible she hid from the guards, or there was a breakdown of procedure at the time of her arrival. The idea that the Nazis were a well run machine is greatly exaggerated.

I have heard no account of a large toilet. The toilets the prisoners used at Auschwitz were pit toilets with wood seats, like camping or military outpost toilets. In Schindler's List, kids hid in them. They didn't have individual stalls, so she may remember sitting next to someone while using the toilet. Or, it's possible she went into hiding and she's remembering something else. Also, there's an urban legend kids used to scare each other with about going to school and using one giant toilet together - it's so ubiquitous, it was in the Simpsons. Maybe her memories are muddled.

Is it possible that your grandmother isn't Jewish? Can you have misunderstood what she told you about Warsaw?

You see, there were two uprisings in Warsaw during the war. One was the Ghetto uprising in April-May 1943. The survivors of this uprising were sent to Treblinka (as had been the case for most of the inhabitants of the ghetto prior to the uprising) where they were killed upon arrival.

The second Warsaw uprising was by the Polish Home Army in August 1944. It had nothing to do with the Jews, as the ghetto had been destroyed and all the Jews removed the year before. Many Poles, including children, were sent from Warsaw to Auschwitz following this revolt. They were not killed upon arrival because only Jews and "Gypsies" (Roma and Sinti) ever were. About 500 children under 15 were liberated by the Soviets when they entered Auschwitz in January 1945, They were left behind as too weak to travel on the death marches, as were about 6,500 adult prisoners.

A quick book rec—Shimon Huberband wrote an account of Warsaw during the war, both before the ghettoization as well as after it, including the system of labor camps. Might be useful information about that side of things. Here’s the link.