If I'm grappled, can I cast Misty Step? What about Dimension Door? What about Thunder Step?

Yes, you can absolutely cast those spells to escape!

Here is how the rules interact:

• The Grappled Condition: Having the Grappled condition reduces your Speed to 0, but it does not restrict your actions, bonus actions, or your ability to cast spells D&D 5e SRD 5.2 — Gam....
• Teleporting Breaks Grapples: Teleportation does not require you to spend your Speed. Both Misty Step and Dimension Door instantly move you to a new location. Because the rules state that a grapple ends when "the distance between the Grappled target and the grappler exceeds the grapple's range" D&D 5e SRD 5.2 — Gam..., teleporting away will instantly free you.
• Thunder Step: While Thunder Step isn't included in the freely available SRD 5.2 rules I have access to, it works the exact same way. If a spell enables you to teleport out of the enemy's reach, it will break the grapple D&D 5e SRD 5.2 — Gam....

(Bonus Note): Both Misty Step and Dimension Door only require Verbal components anyway, meaning you don't even need a free hand to cast them!

This is just a taste of what LorePanic can do — with the full app, GMs can upload their own rulebooks (like Xanathar's Guide to Everything for Thunder Step!) and instantly resolve complex rule interactions for their specific campaign.