>other peoples called them Roman

Depends. Arabs called them Romans because they saw a continuity in the East between Roman and Byzantine rule. In the West, there was no such continuity and medieval Europeans called them just Greeks.

>but latin speakers were always present

By the 10th century the only Latin speakers in the Empire were a few Vlach/Aromanian sheperds in the Balkans.

>latin was till widely used

By the time of Heraclius, Latin was only used in offical military documents, something he got rid of because it was literally just something left from old imperial administration which wasn't useful in an empire where Greek was the majority language as well as the lingua franca.

It was certainly in a terrible state. I can hardly remember the name of a prominent Arabic piece of literature between the Ottoman conquest of the region and the Arab renaissance of the 19th century. Most Arabic texts from the time were religious in nature and this was a result of the decline in urbanism and literacy at the time. The trend technically started ever since the Mongol conquest but there was a small attempt at an Arabic revival by the Mamluks.

The thing is, it wasn't "rescued" by the West. The Arabic renaissance was mainly due to the rise of Mohammed Ali Pasha who was a great patreon of Arabic literature (ironically, despite speaking little to no Arabic). He helped create a network of writers and artists that linked the Levant, Arabian Penisula and North Africa from various religious backgrounds allowing Arabs from various parts of the Arab world to produce great artistic and literary works. His succesors weren't as interested as he was but the network he created continues (arguably) to this day.

When it comes to Western influences, it is undeniable there existed many. Basically, the form of literature that arose was intended to merge medieval Umayyad/Abassid styles with modern Western styles. They saw themselves as a continuation of the Medieval style after centuries of decline which can be seen in their continued use of a modernized and revived form of Classical Arabic (MSA) but they still were inspired heavily by styles from the West after they got exposed to them in their travels. For instance plays and novels entered the Arabic literary tradition at this period.