Hoy necesito la muleta para caminar por la casa.

In Spanish, singular countable nouns usually need a determiner (like el/la/un/una/mi/tu) when they’re the direct object: necesito la muleta / necesito una muleta.
Saying necesito muleta sounds incomplete to most speakers because muleta is a countable item.

• la muleta = the crutch (a specific one the speaker already has/uses regularly, or the only relevant one).
• una muleta = a crutch (not specific; could be any crutch, possibly implying they need to get one).

So Hoy necesito la muleta... often suggests “today I need my/the crutch (the one I usually use).”

• muleta = a crutch (often the underarm type; can also be used for forearm crutches depending on region/context).
• bastón = a cane/walking stick.

In many Latin American contexts, muleta is the normal everyday word for a crutch.

Because para expresses purpose/goal: I need the crutch in order to walk.
Por usually expresses causes, reasons, movement through/around, exchange, duration, etc. So para caminar = “to walk / in order to walk.”

Por la casa is flexible and often means around/through the house, emphasizing movement within the space.
If you want “at home” (location, not movement), you’d more likely use en casa.
So caminar por la casa highlights walking around inside the house.

You can, but it changes the nuance:

• caminar por la casa = walk around/through the house (movement through areas).
• caminar en la casa = walk in the house (more neutral location; less “moving around”).
  Also, for “at home” as a general idea, Spanish often prefers en casa (no article) rather than en la casa, unless you mean a specific house.

Yes. Hoy is flexible:

• Hoy necesito la muleta... (common; sets the time first)
• Necesito la muleta hoy... (also natural; slightly more emphasis on needing it)

Spanish word order is more flexible than English, especially with time expressions.

Necesito is a direct, common way to say I need. It can be strong or neutral depending on context.
If you want softer phrasing, Spanish might use:

• Me hace falta la muleta... (I’m missing/it would help to have the crutch)
• Creo que voy a necesitar la muleta... (I think I’m going to need the crutch)

Because caminar usually isn’t reflexive. You just say caminar = “to walk.”
Reflexive forms like caminarse are uncommon and mean something different in special contexts (often idiomatic), not the basic idea of walking.