How far back do you want to go? In the Bronze Age, Anatolia was mostly under the domain of the Hittites, which spoke an Anatolian branch of Indo-European. Following their collapse, Anatolia was inhabited by various other Indo-European peoples, such as Phrygians, Greeks, and Persians. It was the expansion of the Seljuk Turks that brought Turkish-speakers into Anatolia, and they became the dominant people over the next few centuries. The Mongols never really entered Anatolia--the Seljuks were vassals of the Ilkhanate, but Mongolian-speaking peoples didn't migrate into the region like the Turkic-speaking peoples did.
Further afield, the Central Asian steppes were apparently first settled by Indo-European languages of various branches (Iranian and Tocharian). Alexander the Great's conquest led to the establishment of Greek as a major upper-class language throughout Bactria although it doesn't appear to have transfused down to local populations. Turkic nomads started filtering in and pushing out/assimilating the extant Indo-European speakers in the 400s or so (long before the Mongol Empire).
As for traces of loanwords, yes, many Turkic languages incorporated Indo-European (particularly Iranian/Persian languages) of extant speakers into their language, much like how French incorporated Germanic elements into its language.