The answer to your first question is yes, Auschwitz was the only camp which tattooed identification numbers on its prisoners. Prisoners with number tattoos were liberated from other camps, giving the mistaken impression that they were tattooed at those camps, but in fact those prisoners had been tattooed at Auschwitz and then transferred to other camps.

However, not every Auschwitz prisoner got a tattoo. The practice of tattooing only began with the arrival of Soviet prisoners of war at the camp in October 1941, almost a year and a half after the camp opened. It was expanded to incoming Jewish prisoners in 1942, and then to all other prisoners in 1943. Soviet POWs were tattooed on the left side of their chests, while all other prisoners were tattooed on their left arms. The original purpose was to allow camp personnel to identify dead prisoners after their numbered prisoner uniforms had been removed, but it was later used as a method of identifying escaped prisoners as well. Only a few prisoner groups were exempt from tattooing: ethnic German prisoners, police prisoners, and so-called "labor education" (Arbeitserziehung) prisoners (people who had deserted from forced labor or refused to work were sent to "labor education" camps for eight to ten weeks as punishment).

However, even once tattooing became the default practice for most prisoner groups, some prisoners still didn't get tattoos. People who were selected for the gas chambers on arrival weren't given tattoos, since they weren't registered in the camp and therefore there was no need to identify them, and prisoners who were transferred to other camps shortly after arrival weren't tattooed either, since they were also not registered in Auschwitz. I guess that sort of answers the second part of your first question as well; sometimes prisoners were brought to Auschwitz and then transferred to other camps if they were selected for forced labor, so it's possible that your grandfather spent his entire time in captivity at Auschwitz, but it's also possible he may have been sent elsewhere to work, and the tattoo alone doesn't tell you anything for sure beyond the fact that he spent at least some time at Auschwitz.

The fact that he was a non-Jewish Czech but received a tattoo suggests that he was taken there for forced labor within the Auschwitz complex, rather than as a police or labor-education prisoner, since they weren't tattooed. It's worth remembering that Auschwitz was more than just a concentration and extermination camp. It was also the center of a sprawling system of forced labor camps for both Jewish and non-Jewish laborers, with one main forced labor camp (Auschwitz III, at the Buna synthetic rubber plant in Monowitz/Monowice) and dozens of subcamps, mainly supporting German war-related industries. Most of these camps were located in what's now southern Poland, but there were a few in Bohemia and Moravia as well. Thousands of forced laborers were employed at these camps, particularly during the later years of the war, when Nazi Germany was heavily dependent on foreign forced labor to sustain its war economy. It's possible that he spent his entire time of imprisonment in one of these camps, or he may have been registered at Auschwitz and then transferred to another camp. It's hard to say without more information; all the number tattoo tells you for sure is that he was registered as an Auschwitz prisoner at some point, probably in 1943 or 1944.

As for the treatment he received, it would have depended on exactly why he was sent to the camp. There was a distinct hierarchy within the German concentration camp system, based on the prisoners' nationality, race, and reason for imprisonment, with Germans, particularly those arrested for non-political crimes, at the top and Jews obviously at the bottom. Roma (Gypsies) and Soviet POWs were also treated poorly; fewer than 3,000 of the 23,000 Roma brought to Auschwitz survived, and almost all of the 15,000 Soviet POWs brought to Auschwitz had died by spring 1942, with fewer than 100 surviving until liberation. So-called "asocial" groups like Jehovah's Witnesses and gay men were also low on the hierarchy and were severely mistreated, sometimes even by other prisoners.

Non-Jewish forced laborers fit somewhere in the middle of this hierarchy. They would have received somewhat better treatment than the Jewish prisoners and other groups further down on the hierarchy, but were still forced to live under a harsh regime, working twelve hours a day, six days a week, in factories and farms; living in poorly-constructed, unsanitary barracks; and receiving insufficient food (usually no more than watery soup and a small amount of bread) and medical care that led to widespread malnutrition and disease. It's hard to be more precise without knowing exactly where your grandfather was and why, but the conditions in the forced labor camps within the Auschwitz system were always bad, and got even worse toward the end of the war as the strain on Nazi Germany's resources increased. Even though the non-Jewish laborers weren't the target of a program of industrialized mass-murder, they were still treated very badly, and more than 10,000 of them died in Auschwitz and its various subcamps.

Sources:

Wacław Długoborski and Franciszek Piper, eds., Auschwitz, 1940-1945: Central Issues in the History of the Camp (Auschwitz-Birkenau State Museum, 2000), 5 vols.

Debórah Dwork and Robert Jan Van Pelt, Auschwitz (W. W. Norton, 2002)

In addition, the USHMM Holocaust Encyclopedia has an article on prisoner tattoos that might be of interest to you.