I can say that the right word, as far as culinary art is concerned, appears nowhere in your question. The term for “hot” would be more specifically “piquant”: a hot spicy food → de la nourriture fortement épicée; one is never too sure what is meant by “épicé”: it can be “piquant” (that is stinging on the tongue and palate) or it can mean that it has strong spices, such as curry that isn’t spicy in the way that hurts your mouth.

For the time being, “chaud” is to be banished from your French vocabulary in the way of rendering the adjective “hot”, except in the context of actual heat, as corrected @Luke Sawczak in the comments.

• the electric line is hot → “il y a le (du) courant” ou (substandard) “il y a le jus”
• hot news → nouvelles de première page

However, you might — before long if not already — be able to translate “hotline” by “ligne chaude” and similarly for a few others. “Hotline” is now part of the French vocabulary.

In French, is it valid to use “chaud” to refer to hotness of chillis (i.e. the burning sensation and not temperature)?

It is more than unlikely to be used. Chaud essentially relates to the food temperature when applying to dishes. However, we might say en feu to describe the feeling: avoir la bouche en feu.

Is there a one-to-one equivalent translation of “hot and spicy”?

We use fort (strong, opposed to doux: sweet/soft) for hot food like in moutarde forte/douce or piment fort/doux so “hot and spicy chillies” would be piments forts et épicés.

We also have an adjective that encompass both hot and spicy when referring to a dish: relevé

Ce plat est (très) relevé.

Note that while picante is the main word used to translate hot (chillies) in Spanish (as you already know) and fuerte a much less used alternative, this is the opposite with the French corresponding words piquant and fort.

Spanish:

French:

On the other hand, when qualifying sauces, piquantes is leading in French too:

Pour un plat piquant à cause d’un piment, on peut aussi utiliser l’adjectif “pimenté”, tout simplement. L’adjectif “relevé” est aussi une solution. Par contre, je confirme que “chaud” ne s’emploie pas dans ce contexte.

En ce qui concerne la température, il me semble que “hot” correspond à une température élevée (mais je ne suis pas anglophone). Je traduirais donc par “brûlant” (pour un plat solide) ou “bouillant” (pour un plat liquide) si le but est de prévenir une brûlure de la langue en mangeant.

Shame on me for thinking it necessary to offer some colorful (read: “arguably gutter-talk”) options to go with all the excellent answers given so far, but the often not-so-nice (when applied to humans) word gueule (= la bouche= “mouth”), figures in several descriptive French expressions I’ve heard/used involving spicy hot foods (and the strong/harsh alcohols of my youth).

Ça/Cela arrache la
gueule =
c’est très pimenté = “That’ll take the roof of your mouth off!”
(from WordReference.com), …

from which (according to ‘viera’ in this WordReference thread) the adjective …

arrache-gueule is [apparently derived and] commonly used as an adjective to refer to very hot (spicy) food.
“Cette sauce est plutôt
arrache-gueule.”

Although more for describing one’s ability to tolerate/stomach/enjoy such food than for describing the food itself, from the French Academy’s Dictionary entry for “gueule” (at the link provided in paragraph 1 above, via CNRTL), there’s:

Avoir la gueule pavée = être capable d’avaler des aliments très épicés ou très chauds.*

And according to this mediadico.com entry for “gueule,

Avoir la gueule ferrée can have the same meaning as the above expression using pavée.

(both of which could translate to “Having mouths/pie holes/gullets lined with steel” in English)