No Program is a Substitute for Reading the Book
While using a character generator is great for making characters faster, it is no substitute for reading the book. Especially with all the wonkiness that you can get in Traveller chargen!
If you succeed an advancement roll, you get an extra roll on the Skills and Training Tables for the career you are currently in (pg. 16–17) and you also gain a rank. Some ranks have bonus skills in addition to the rolls you get on the table.
Note that each term also nets you one roll on the Skills and Training Tables. This means that when you succeed on an advancement roll, you gain another roll, and a possible third skill if your new rank gives you one!
as a Rogue / Enforcer, on my first advancement roll I thought I should get Persuade 1, and that is the end of it. But the tool gives me both Persuade 1, and a choice to roll on one of the three tables Personal Development, Service Skill and Enforcer, in effect giving me Persuade and another skill. Is this correct or should I only be given persuade?
On your first term as a rogue/enforcer, you get to roll once on one of the following tables of your choice: Personal Development, Service Skill or Enforcer (if your EDU is 10+ you can also choose Advanced Education).
If you succeed on your advancement roll, you get to roll again on any of the appropriate tables. And, since you advanced to Rank 1, you also get Persuade 1.
Same example as above with being promoted to Enforcer rank 1. What happens if I already possess Persuade 1? Does it increase to Persuade 2 or is that benefit lost?
Per page 16, "If a rank is listed, then you gain the skill at that level so long as it is higher than your current level in that skill. For example,…Streetwise 1 would mean you get the Streetwise skill at level 1. [i]f your Streetwise skill is already 1 or more, then this is of no benefit to you."
In your case, if you already had Persuade 1, you would gain no benefit.
Some careers have a skill listed for Rank 0, like Marine that has "Gun Combat (any) 1 or Melee (blade) 1". Does this mean that once you enter this career you get this skill in addition to Basic Training?
Page 17 tells us that "Some ranks have bonuses associated with them, such as extra skills. These are acquired immediately upon attaining the rank." Your first rank in any career is rank 0, so you gain any skills associated with rank 0. In the case of the marine enlisted, at rank 0 you would indeed gain the listed skills.