I can only answer to the lamp shade part of the question.

There are two believable sources that one such shade actually existed. Dr. Gustav Wegerer, who was brought to the KZ Buchenwald for political reasons and was a KAPO in the pathology, declared in lieu of an oath, that the Lagerkommandant Koch and a doctor named Müller came to him while they were preparing a lamp shade out of tannened and tatooed skin. They both picked skins that were to their liking for this out of a number of skins that were available. Overhearing their talk, he assumes that they had to choose different skins, because the ones used before didn't please Koch's wife.

Josef Ackermann, who was also incarcerated for political reasons and secretary to Waldemar Hoven, the camp doctor, later said in front of a judge, that he was chosen to deliver the lamp to Koch on the occasion of his birthday in August 1941. He claims that the foot of the lamp was made out of a human foot and shin. The shade was indeed out of human skin, and nipples and tattos were visible. One of the party guests later told him, the lamp was supposedly a huge success with the crowd attending the party. But, after higher ranks of the SS had gotten wind of the lamp, it had to disappear.

Koch was arrested at the end of 1941 due to corruption and dissolute furthering of power ("zügellose Machtentfaltung", I have no idea how to translate this correctly). There are certain pointers that the shade was destroyed in this context.

After the liberation of Buchenwald, on April 16th, 1945, the US Army presented a certain number of objects from the pathology, among them tanned human skins, two shrunken heads and a lamp shade. This can be seen on pictures that were taken at the time (I'm not linking them, google them if you want to see them).

Nevertheless, the lamp shade was not among the objects that were tested for authenticity in May that year. It is unclear what happened to it and what material it was made from. Dr. Harry Stein assumes that it was a lamp from the office of Hermann Pister, who was Lagerkommandant at the time of the liberation.

The human skins were brought to archives in the U.S., though, one of them, which is now in the National Archives, is cut in form of a trapeze and has holes in it, so it was clearly intended for processing.

Edit: I tried to find an English source, but didn't manage to, I'm afraid. You'll find Ackermann's statement (paraphrased) in Die Hexe von Buchenwald: der Fall Ilse Koch by Arthur L. Smith on page 192, parts of the statement (paraphrased) by Wegerer on page 131. The whole statement (verbatim) by Wegerer can be found in a short statement by Dr. Harry Klein, Kustos of the museum at Buchenwald under http://www.buchenwald.de/1132/

Since the part about lampshades has been already answered, here is the info on the soap:

Yes, the Nazis did indeed produce soap from the human fat of Concentration Camp victims. This has been confirmed in 2006, although it needs to stressed that they never did it on an industrial scale but only in small quantities and it was experimental.

The production of this kind of soap was located in the Stutthof Concentration Camp near Danzig/Gdansk by a man named Rudolf Spanner of the Danzig Anatomical Institute. An assistant of Spanner testified at the Nuremberg Trial that fat from about 40 people was collected to make about 25kg of soap. The use of that soap remains unknown as the produced bars have according to the lab assistant's testimony been retained by Spanner.

It is important to stress here that what has been general accepted as evidence - bars of soap labelled "RIF" - show no trace of human DNA and the RIF stands for "Reich Industrial Fat" and does not pertain to Jews.

Sources:

• Press releases of the USHMM and the Auschwitz Memorial Site.

• Joachim Neander: „Seife aus Judenfett“, paper presented at the Oct. 2004 German Studies Association conference.

The only example I can give is about Franz Ziereis and his commandants under is supervision. Franz Ziereis was an SS camp commandant who was know for his brutality. "Other commandants at subcamps under his command tanned the tattooed skin of victims' bodies for use of bookbindings, lampshades, and leather satchels. Ziereis refrained only because Berlin quickly forbade the practice."
I do only have one source pertaining towards this practice. The book is titled, "Becoming Evil: How Ordinary People Commit Genocide and Mass Killings." I highly recommend this reading for further detail.

As a follow up question, its true that the Nazis used human hair from camps to stuff mattresses?