Does scaling apply if I don't level up?

No.

Once you reach the next level, you do not gain a level for scaling purposes until you have taken that extra boost to Health/Magicka/Stamina. However, since you said "saving my perk points," keep in mind that you are not required to pick a new perk every time you level up; you can save them until you meet a particular requirement. In fact, it's actually quite common for perk points to exceed desirable perks if you have a particular build in mind.

In my opinion, it's generally not worth saving a level up for a few levels just so you can, say, take four perks you want all at once. The exception I can think of is if the intermediate perks are really awful and you're struggling with difficulty. Barring that, I prefer either taking the level up at once or using it opportunistically (you get a complete refill of health, magicka and stamina), especially early game.

I do not think so.

Although they continue to accrue through more skill use, the level ups are only applied when you open your char menu and assign the compulsory stat point. Before that point you have not levelled. Assigning the perk point is not necessary for a level-up.

All level related things are on hold until this point including your level, training points and level-dependent quests.