Siege towers were definitely a real thing, but they were certainly not used at every siege, or even most. My familiarity with medieval sieges has more of an emphasis on Crusading warfare so I'll be focusing on that for my examples. If you want a good general work on medieval sieges you should check out Jim Bradbury's The Medieval Siege.

Two famous Crusading sieges featured the use of siege towers: The Siege of Jerusalem in 1099 and the Siege of Acre in 1189-91.

The Siege of Jerusalem sort of marked the end of the First Crusade. I say sort of because while the taking of Jerusalem marked the Crusaders achieving their main goal the Battle of Ascalon was pretty essential to ensuring that they didn't immediately lose it to the Fatimids. The Crusaders succeeded in taking Jerusalem with the help of two siege towers they built from whatever wood they could harvest around the city. Their army was too small to encircle the city and besides that they didn't really have time to wait and risk the Fatimid army from Egypt coming and attacking them outside the city. Instead they built siege towers and the one lead by Godfrey of Boiullon managed to reach the walls and from there they opened the gates and took the city.

The Siege of Acre was a really long siege fought during the Third Crusade where Richard I eventually succeeded in taking the city. The siege tower story is a bit shorter here. The crusaders tried to take the city with three siege towers (I can't remember which year off the top of my head) but the Muslim defenders succeeded in destroying all three before they reached the walls (usually by lighting them on fire). The Siege of Acre was eventually won by starving the city until it surrendered.

For Crusading history in general I like Thomas Asbridge's The Crusades. For a more specific account of the Siege of Jerusalem you should check out Conor Kostick's The Siege of Jerusalem.

If you want to read a period source that describes some siege weapons and their use, you can read the medieval account On the Conquest of Lisbon, written by a crusader that helped in the 1147 siege.

Among other things he describes the construction and use of a 83 foot (25 meter) siege tower. It is interesting to notice that /u/valkine's answer refers mainly to the crusades, and so does this, since the crusaders who built the tower were on the way to the Middle East and stopped by just to give a hand to the Portuguese king D. Afonso Henriques (and maybe get some extras on the side if you get my drift)

The full English text is available here http://catalog.hathitrust.org/Record/000632415, but just for the sake of anyone not wishing to read it all (I would recommend it to anyone with interest on the topic of medieval warfare) I will place some choice quotes here.

They start building it

And when we had been at the siege for a fortnight, we began on both sides to build engines—the men of Cologne and the Flemings constructing a sow, a ram, and a movable tower, and we a movable tower ninety-five feet in height.

At last our tower was completed and covered all around with collision mats of osiers and with oxhides in order that it might not be damaged by fire or by the impact of stones. Thereupon, an order was issued by proclamation throughout the fleet that all should make penthouses and mantlets of interwoven branches. And so, on the following Sunday, when everything necessary for its defense had been put in position, the archbishop was sent for in order that the tower might be moved forward with his benediction.

The tower is in use at last

Again next daythe engine was moved up towards the tower of the city which stands at the corner towards the river; to which also the enemy had brought the full garrison for its defense. When this was discovered, their plans were easily defeated; for our men, turning the engine to the right towards the river, passed the tower at a distance of about twenty cubits and brought it near the wall almost at the Porta do Ferro which overlooks the tower.

The moors counter attack

Then during the first watch of the night the incoming tide flowed around our engine and cut our forces off from communication with it. When the Moors observed that the rising tide had cut us off, they issued in two companies through the aforesaid gate and attacked our machine.And the others upon the wall, an incredible multitude, brought up articles of wood, together with pitch and flax and oil and every kind of inflammable matter, and hurled them at our engine.

Then the crusaders respond

And under this some of our men, with the assistance of those who were inside the engine, beat out the burning matter piece by piece as best they could. And others dug trenches underneath the engine and, lying in them, drew away the balls of fire. Still others in the upper stories poured water through holes upon the hides which were suspended from above so that the tails hanging in order on the outside irrigated the whole engine.

This goes on for some pages, and in the end the tower ends in flames. As said before if you like it read the full thing, this section starts around page 160 in the English source provided above.

Alexander built two towers and a causeway at Tyre, which ended up burning down, though he eventually took the city. He used towers at Halicarnassus also. You get can read about them in Arrian's Anabasis.

Caesar used towers at Avaricum. He even built ramps, because the towers weren't tall enough to do the job. He gives a pretty vivid account of that siege. The Gauls undermined his ramp and built higher walls while he was building his tower. There are night time raids, towers on fire, special constructed walls.... it's an interesting read. That's in Caesar's Gallic Wars book 7.

Somewhat related: do we have any partially/fully preserved siege towers, or at least designs from the period to allow for reconstruction? Or is the bulk of our knowledge from art and writings?

I think an interesting takeaway that you might like is that siege warfare (like all other types of warfare) developed depending on the environment. For example, in Japan, siege towers were never used, for a couple reasons.

'Castles' of Japan weren't really the type of castles you see in Europe, not until much later - for most of the 'Middle Ages' period in Japan, castles were essentially fortified outposts normally situated on mountains or hilltops, but didn't house an entire family or town as you'd expect it to. They were used simply as military 'strongpoints' or as signalling beacons from one region to the next. Temporary military fortifications were often built with a proper escape plan in mind, in case things went bad - this is probably another reason why siege warfare was more shaped around trebuchets and arrows to inflict damage on soldiers rather than use a siege tower.

Also, due to the nature of Japanese warfare in the periods of civil war, castles built later on in the Sengoku period were more focused on choke points, or large winding paths for enemy invaders to get to. If you wanted to reach the central keep, you'd have to build a siege tower for the first wall, get over that wall (whilst having arrows and all sorts of stuff rained on you), then somehow magic up some more siege tower materials to get past the second wall (remember, you still have stuff being shot at you) - just not worth the effort, when you could probably try first shelling the castle with cannonballs or trebuchets, or something at the very least more effective than dedicating time, effort, manpower, and lives on siege towers.

There is an actually ancient illustration of an Assyrian siege engine from the Assyrian palace at Nimrud.

How has there been no mention yet of the Helepolis that Demetrius Poliorcetes used at the Siege of Rhodes?