I’ll answer in order! See below th questions.
For the initial mission, each player creates one character without any restrictions. Also, there’s a squad of rookies (5 rookies) coming along the main characters, right? Theoretically, all squads are at full capacity at this time.
You got that right. And the Marshall completes the number of Specialists to 5. For instance, if you have 3 players, they each create a Specialist, and the Marshall creates 2. There’s no obligation to have different playbooks for all those 5 Specialists (at least, it’s not written in the rules), but of course it’s better (or? it dépends on how you wanna play). However: it’s a bit strange, but even if each player creates a Specialist, they don’t have to play those on the first mission. They can play a Rookie, or they can play a Specialist created by another player.
The initial 5 extra rookies belong to one of the squads, so if they die (or if one of the players creates a soldier or a rookie), that squad will lose members?
Yes. And if the Legion finds itself with less than 15 Rookies (equivalent to 3 full squads), the game is lost. Same if you don’t have at least 2 Specialists alive.
Specialists do not belong to any squad, squads can only have soldiers and rookies. If a player soldier advances to specialist, they stop belonging to that squad.
Squads have a maximum of 5 members each.
When can new specialists be created after the first mission? If there’s no restriction to creating specialists, players could always create a full party of them for each mission?
You don’t create Specialists after the first mission. Either you promote them through the ranks (it will take a bit of time), or you get some specialists from some mission rewards. This means that you can find yourself with a shortage of Specialists, and especially with a shortage of Specialists able to go on a mission.
Also: you cannot have a “full party of Specialists”. Except for a very few exceptions (the first mission is one), you are not allowed more than 2 Specialists on any mission, or 3 if one of them is a Medic. If you have 3 or more players, some of them will have to play squad members. Same if your specialists are too heavily wounded.