No, a cleric doesn’t need to worship a god. That’s categorically false. Paladins in 5e do not need to follow a deity, but Clerics absolutely do.
Can clerics speak to their gods?
An Arcana Cleric would more likely rely on periodic rituals to communicate with their deity. A divination ritual involving candle wax, special playing cards or inkblots seem appropriate. A Grave Cleric may have some practice similar to the Twilight Cleric you described.
Can you be both a cleric and a warlock?
Spell slots would be kind of a pain, but I think it’d be neat to be both a divine and arcane caster, and have both healing magic and attack magic be equally powerful. Asmodeus is also a classic example; he’s the epitome of both lawful evil deities and infernal warlock patrons. In 5E, his clerics have access to the Knowledge and Trickery domains.
Can a Divine Soul Sorcerer be a cleric?
Go with divine soul Sorcerer instead. You get full access to the cleric spell list as a Chasima caster. You could flavor your Sorcerer ability as a gift from god…making that work with your patron could be tricky though. You’re not necessarily in it for the Cleric spells; sometimes it’s the low level domain powers that count.
Do you need charisma and wisdom to be a cleric?
The obvious problem you’d run into is with your stats. You’d need both high Charisma and high Wisdom. And since your two class features that let you cast spells wouldn’t stack, spell slot allocation would suffer. Go with divine soul Sorcerer instead. You get full access to the cleric spell list as a Chasima caster.
Can a warlock be both a demon and a goddess?
Lolth counts as both a demon and a goddess, Ghaunadar is considered a War domain god for clerics and a Great Old One patron… There are lots of cases where gods could also be patrons. Also, having high Charisma and Wisdom is possible– I have a paladin running whose two highest stats are Cha and Wis.