This map is from Wikipedia's Varieties of Arabic page. On the northern half of the western Red Sea coast, as well as most of Sinai and Israel, it seems to indicate that "Badawi" Arabic is spoken, but I can't find for the life of me any sources on the topic. My questions:
This is intended to refer to the dialects of the bedouin communities of eastern Egypt, the Sinai and the Negev desert. It is most closely related to the bedouin dialects of Jordan and northwestern Saudi Arabia and is the same as what Wikipedia refers to as "Northwestern Arabian Arabic". It is closer to what you would hear in the Jordanian and north Saudi Arabian deserts than what you would hear in Cairo in terms of phonology and vocabulary, but not as conservative as your typical Arabian bedouin dialect.
No, this map is not correct. It was recently removed from Wikipedia because it has numerous errors & is largely unsourced. With regard to "Badawi Arabic", this map collapses a historically bedouin version of Arabic spoken in the Levant & Sinai peninsula with a superficially similarly named Cushitic language "Bidhaawyeet" spoken mostly in the Sudan. Most of the area along the Red Sea Coast in this map comes from sources on Bidhaawyeet. (Arabic is also spoken in these areas, but it's not the "Badawi Arabic" in question.) As a starting point for linguistic sources, I think Glottolog is usually good. Here's their Eastern Egyptian Bedawi Arabic page: https://glottolog.org/resource/languoid/id/east2690 De Jong's grammars are a good starting point.