[RPG] Can a character spend multiple hit dice at level 1

The relevant quote is this on page 15 of the PHB, or here in the basic rules:

When your character gains a level, his or her class often grants additional features, as detailed in the class description.

Nowhere does it mention that it takes any time to gain these features when you level. The only exception is that it explicitly states that leveling increases the hit point maximum. Since 5e defines current hit points and maximum hit points as specific terms, you are right to conclude that the you don't gain your new hit points until you rest. However 5e doesn't make that distinction in the terms of hit dice. Like your proficiency bonus, attribute gains, etc., it takes effect immediately upon leveling.

It is reinforced by the fact that Hit Dice advancement is in its own sentence:

Each time you gain a level, you gain 1 additional Hit Die.

Then the following sentence states:

Roll that Hit Die, add your Constitution modifier to the roll, and add the total to your hit point maximum.

Which make it a separate point.

Number 3. From page 163 of the PHB:

You add together the Hit Dice granted by all your classes to form your pool of Hit Dice.

[...]

If your classes give you Hit Dice of different types, keep track of them separately. If you are a paladin 5/cleric 5, for example, you have five d10 Hit Dice and five d8 Hit Dice.