"Senior", "Junior", and similar identifiers are formal parts of a person's name in any English-speaking country.
Most English-speaking countries don't have identity cards. As for the passport, that would be at the discretion of that country's passport agency, but my guess would be typically yes.
Generally, when you read last name it really means family name or sur name.
But what do you mean when you say "family name"? Are your siblings also called ___ Adel Mahmoud Gad? Or just given-name some-other-name Gad?
If your whole family uses the names Adel Mahmoud Gad, you should use that as your "last name." I'm not sure about the rules in your culture, but many Spanish and Hispanic people have multiple family names, for example Gabriel García Márquez. His "last name" is not Márquez, it's García Márquez.
If you're working with a webform that's too stupid to accept spaces in a last name...I guess just use Gad.
As for middle names, fortunately, these are rarely required. If you decide that Gad is your family name, you could use both Adel and Mahmoud as your middle name. Again, if it's a webform, and inflexible, you'll just have to pick one. For official documents, it doesn't really matter what you use as a middle name, but just be consistent.
Best Answer
No; junior and senior are only used if the names are exactly the same.
Technically, a child named after a relative who is not his father should be called "Jack Watson II" (pronounced "Jack Watson the second") instead of "junior". Then if there is a third Jack Watson in the family, he would be "Jack Watson III" ("Jack Watson the third"), and so forth.
(I actually did once know a person who was called "John Smith junior" even though the original John Smith was his grandfather and not his father, but this was unusual enough that people remarked on it, because it is not the normal way to do it.)
Also, junior or senior is almost always written as Jr. or Sr. instead of being spelled out.
Source: The Emily Post Institute, "Men's Names and Titles"