I do not know if I understood your questions correctly, but let me try to help:
- Some spells improve with level, especially cantrips get better at lv. 5, 11 and 17.
But what level? Character level, level of the class that gave you this spell or something else?
When multi classing, does it matter if the other class has the same spell too, is a caster class or cannot cast spells at all?
As per PHB:
"If a cantrip of yours increases in power at higher levels, the increase is based on your character level, not your level in a particular class", that includes cantrips you get from feats, race, class, etc.
"Each spells you know and prepare is associated with one of your Classes, and you use the Spellcasting Ability of that class when you cast the spell. Similarly, a Spellcasting focus, such as a holy symbol, can be used only for the Spells from the class associated with that focus" So if you learnt fire bolt with your wizard class you will use INT, if you learnt it with your sorcerer class you will use CHA and in D&D 5e almost all the spells (not cantrips) only get better if you use them with a higher spell slot.
i.e) if you want to shoot four magic missiles instead of three does not matter if you have 3 levels of wizard or 20 levels, you still have to use a 2nd level spell slot, generally speaking.
If you learn the cantrip from other sources, like feats, they tell you in the description what stat you use for spellcasting, usually. So no, whatever the other classes you multiclass with, they do not pile up. But since the cantrips are heightened with your character level, it does not matter for cantrips.
- All spells from different classes share the same spell slots.
So a sorcerer5/cleric5 has the same amount of spell slots like a single class character, but only half the number of different spells per level.
Paladins and rangers count as 1/2, fighters and rogues with spells as 1/3 when you multi class with a full caster.
+ How does it work when you combine those classes (paladin, eldritch knight and so on) when not having a level in a full caster class?
+ How does it work if you combine a caster class with a warlock, who has a completely different spell system (only a few slots, you use always the highest slot level, only slots up to lv5 higher level spells can only be used once per day, spell slots recover at short rest)
In short, you just follow the multiclass table of WoTC
https://5thsrd.org/rules/multiclassing/As you said, Pally and rangers count as 1/2 and rogues with 1/3 when you multiclassing to calculate your spell slots, same with each other. With a level 6 character in mind:
So a ranger (3) pally (3) has [4 level-1 spell slots, 2 level-2 slots].
A ranger (3) rogue (3) has only [3 Lvl-1 spellslots]
A ranger (3) cleric (3) has [4 level-1 spell slots and 3 level-2 spell slots]
A cleric(3) druid (3) has [4 Lvl-1; 3 lvl-2, 3 lvl-3]
If you roll a barbarian-ranger you use the spell progression of the ranger class.
For warlocks:
"Pact Magic: If you have both the Spellcasting class feature and the Pact Magic class feature from the Warlock class, you can use the Spell Slots you gain from the Pact Magic feature to cast Spells you know or have prepared from Classes with the Spellcasting class feature, and you can use the Spell Slots you gain from the Spellcasting class feature to cast Warlock Spells you know"i.e. The spells slots of the warlock and any other class pile up with each other, so you can use either.
A ranger (3) warlock (3) has 3 level-1 slot spells and 2 level-2 slot spells from the warlock class and so on...
- You cannot cast spells with somatic component if you have a shield or a 2h weapon equipped unless you have the war caster feat. Is that correct?
You need
a free hand, so you can have a shield in one hand and cast with the other, for example.
Somatic (S)
Spellcasting gestures might include a forceful gesticulation or an intricate set of gestures. If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
Material (M) [...] A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus--but it can be the same hand that he or she uses to perform somatic components.Some subclasses or classes or spells have special ways to wield weapons and casting, like sword bards that use the weapon as a spellcasting focus, so they do cast the spell using the weapon in his hand.
SC: Players Handbook, GM guide.