Characters won’t level up no matter how much I play .

You increase the star cards level by unlocking more Star Cards, whether that is from the loot crates you purchase with credits, or by crafting them using crafting parts.

Class Level is in no way affected by what you play in game. You can dump 10 hours into running assault, but it won't level up unless you get the star cards for it.

The only thing that is affected by playing different classes is the additional weapons and attachments that you can unlock by getting kills as that class or by using specific unlocked weapons.

So the experience meter from our Heroes won't go up unless I got a level 3 starcard in the first slot? Only then I'll be able to get to lvl 5 for my second star card? Because at the moment I can't advance to the second starcard slot.

No, not necessarily by getting a level 3 card... you can do it by crafting cards as well. Build up crafting parts by, for instance, completing the 250 arcade kill challenge in each trooper class (that's I think 25 or 35 crafting parts for each one of 4 achievements) and then craft a few cards for your assault trooper. This will increase the card level and unlock your next slots. Similarly, in your daily crate, when you receive star cards, that will increase your card level for whatever class you were in. Increased card level + increased career level will let you upgrade cards.