Here’s what the spell says:
The target’s game statistics, including mental ability scores, are replaced by the statistics of the new form. It retains its alignment and personality.
What counts as a “game statistic”? That term is never defined, so it’s up to the DM to define it.
(The 5e rules have a lot of these ambiguities, so the DM will have to get used to making these rulings as they come up.)
There’s no consistent ruling about whether “game statistics” include equipment (see this question). There’s also not a consistent ruling about whether “game statistics” include immunities like “immune to charmed” or abilities like “angelic weapons” or actions like “healing touch”.
All of this is up to the DM and can vary from table to table.
You’ve also asked: “if I transform myself permanently, can I level up and progress?”
If you permanently transform into a planetar, the planetar does not have the ability to level up. The character that you used to be might still be gaining experience points, or they might not. (This, too, is up to your DM.) But the planetar won’t ever get better.