As the the response by u/idjet presents the general profile of the Ahnenerbe, I will attempt to address the remaining issue present in the question.

Although Research Society for the Studies of Cultural History, German Ancestor Legacy (ger. Studiengesellschaft für Geistesurgeschichte‚ Deutsches Ahnenerbe e.V.), since 1938 known simply as 'Ahnenerbe e.V.' was a political body created by under the auspices of Heinrich Himmler for the largely pseudoscientific research based on various mythological and historical sources, as well as to contribute international data for the racial studies, it was not uncommon for the actual scientists to join its ranks. This was done primarily because in the late 1930s, when the Germany became a de facto totalitarian state dominated by monopartisan NSDAP government, affiliation with a state agency, especially one closely associated with the highest ranking officials was possibly the only effective way to secure grants and gain a possibility for mounting an expedition to a distant areas, that was very generally a costly affair, especially to the territories governed by countries Germany had strained relations with. Such agreements are sometimes likened to the ones made between medieval alchemists who were funded by nobility promising them gold made of lead or similar lofty achievements, although they did it simply to secure means to conduct their own experiments.

A good example of such researchers is Ernst Schäfer, zoologist and geologist who, fascinated by Himalaya region, took part in the 1930 international expedition to Western China and Tibet organized by an American Brooke Dolan, with whom he cooperated in another expedition 1934-1936, after his graduation (Schäfer was only 20 during the first journey). The subject of the research brought the attention of Himmler who proposed that Ahnenerbe shall equip and fund the German expedition to Tibet (such a proposal is often interpreted as an attempt to attract actual scientists and transform the organization into a more academic society). Schäfer accepted the opportunity, although he was very critical towards pseudo-scientific theories Himmler wanted him to pursue, such as the Hörbiger's 'World Ice Theory' (ger. Welteislehre) according to which Atlantis has been destroyed in a gigantic flood caused by the impact of the ice asteroid and the remnants of the Atlanteans created the new civilization in Asian mountains, with the alleged capital in the city of Urbe. Schäfer and his colleagues, interested in the scientific description of geology, flora and fauna of Tibet opposed such ideas and protested the inclusion of a pseudo-scientist Edmund Kiss into the expedition. This drove a wedge between the Ahnenerbe and Schäfer, although the latter needed to make some concessions to actually proceed with his goals.

Now, regarding the second part of the question, it is largely or even completely product of fiction. 'Raiders of the Lost Ark' 'Indiana Jones and the Last Crusade' are set in 1936 and 1938 respectively, with the bulk of the action taking place in the North Africa and the Middle East, i.e. territories controlled by Great Britain. In such circumstances, heavy presence of German soldiers was next to impossible as evidenced by the aforementioned Schäfer's expedition to Tibet. Although the relations between Germany and other countries were not as strained as they became after annexation of Austria and part of Czechoslovakia in 1938, the mere association of the German scientists with Nazi organizations, flaunted by German press (e.g. Nazi political newspaper 'Völkischer Beobachter') could have became a major obstacle. Aforementioned Schäfer and his people experienced it first-handedly, when the German articles presenting them as expedition under the auspices of SS have risen suspicions resulting in their portrayal as 'German invaders' in the local Indian press, what promptly resulted in refusal to enter both Sikkim and Tibet. This has been openly criticized by German consul general in India, Maximillian Count von Podewils-Dürnitz who stated that:

I attribute the refusal primarily to the presentation of the expedition as the endeavour of the SS. It is a known fact that the consider SS to be a policing and intelligence organization and this undoubtedly caused them to treat the scientific goals of the expedition as a mere pretense to the political activity. The article 'Expedition into the uncharted regions of Tibet backed by SS-Reichsführer and Völkischer Beobachter' published in Völkischer Beobachter on 20th April 1938 was regretfully unhelpful, as was the leterhead Ernst Schäfer's German Expedition under the auspices of SS-Reichsführer Heinrich Himmler and the Ahnenerbe Society e.V. in Berlin used prior to the expeditions's departure. Of course, such information became immediately known and caused the English to become reasonably suspicious, with both the London Times and local press publishing notes pointing to association of the expedition with SS.

What is worth noting, Heinrich Himmler understood the problem and addressed his good acquaintance, Adm. Barry Domville directly, asking him for help. The intervention of the latter allowed German scientists to enter Tibet, but did little to defuse suspicion concerning the honesty of the Germans. Thus, given that the mere association of the scientific expedition with the SS caused great problems for the civilians to enter the country, one can easily assume that the German military presence in the Middle East was completely out of question. This could have possibly happened during the war, but again, it must have been limited to the territory of Germany or the areas Germans or their allies managed to occupy, as any organized German presence in the territory of the state that was in the state of war with the latter would have been treated as a military aggression or an act of espionage. And although it was possible if risky to smuggle some soldiers dressed as civilians, the usage of the entire companies of soldiers along with tanks and airplanes was simply impossible.

Engelhardt, I., Nazis of Tibet: A Twentieth Century Myth, in: Esposito, M., (Ed.), Images of Tibet in the 19th and 20th Centuries. École française d’Extrême-Orient - Études thématiques 22, vol. I, Paris 2008, pp. 63-96.

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