Passive abilities equal your level + 3 active abilities at once if I remember correctly.
The number of passive abilities you can have equipped at 1 time, depends on your level. This does not necessarily mean that it is equal to your level.
Instead of looking at it as "class specific" and "universal," look at it like "active" and "passive."
You can have up to 3 active abilities equipped at one time, in addition to whatever the amount of passives are (which changes depending on your lvl)
In your example of laz nade, health buff and self revive, they are all active abilities, (abilities you need to trigger to use) so if you equipped them all, they would fill all your slots, leaving only room for passive abilities (which do not take up an ability slot)