Can a NFL team win their division games and have the best record for games within the division, however not have the best overall record, and be the division champion?
No, this is not possible.
Currently in the NFL, the standings within a division are based on overall record, not the record within the division. So a team losing every division game (0-6 in division) but winning every other game (10-6 overall) would win the division and enter the playoffs over another team that won every division game (6-0 in division) but lost all others (6-10 overall).
Division games do have more weight in the tiebreakers: The first two tie breakers are head-to-head matchups and division record. So if two teams in the division finished 10-6 overall, but one team won all division games and the other lost all of them, the team that won the division games would be the division champ and enter the playoffs.
Yes, easily. A team could go 6-0 against the other teams in its division but 0-10 against every opponent outside its division, thus giving it a record of 6-10. Another team in the division could go 0-6 inside the division but 10-0 outside the division, giving it a record of 10-6.
Does this happen? Yes, but probably not to that level. In the 2017 AFC South, Tennessee went 5-1 within the division and 4-6 outside it, given them an overall record of 9-7 while Jacksonville went 4-2 within the division and 6-4 outside it, giving them a record of 10-6.