Here we use the slang term “rien à fiche” not the first meaning of the verb “ficher“.

It’s an idiomatic form (that feel like old sailor speaking a vulgar bad french) so it is really “fiche” even though as you said it should be “ficher”. We can conjecture that it’s in order sound closer to “foutre”. You also had “Fichtre!” (interjection) that is a mix of both and is equally antiquated.

You could also use tons of alternatives all more or less vulgar.

If you want to be formal use “importer” (cela m’importe peu) ou “attacher/accorder de l’importance” (je n’y attache pas d’importance), ou “être égal” (ça m’est égal). If you want to be ridiculousely formal you could say “Peu me chaut”.

When used as an euphemism for foutre, “ficher” has an irregular infinitive (fiche instead of ficher) and an irregular past participle (fichu instead of fiché).

This is generally explained by the influence of foutre (past particile foutu). This is satisfactory for the participle, but I’ve always found the form of the infinitive a bit baffling (why not fichre or fichtre? – that last one exists but only as an old fashioned interjection, like “golly” or “by Jove”, as mentioned in Jylo’s answer). It looks like the 19th century innovators of fiche interpreted foutre as a stem /futR/ + a suffix /ə/, and backformed fiche (/fiʃ/ + /ə/) from it. I have to accept it as the only satisfactory explanation I’ve encountered, but I must confess it’s a process that seems very unintuitive from my 21st century viewpoint and with my final /ə/ dropping dialect.

Your teacher was thus indeed correct when they offered “Qu’est-ce que j’en ai à fiche ?” as a bowdlerized version of “Qu’est-ce que j’en ai à foutre ?” What’s more, the regularized version “*Qu’est-ce que j’en ai à ficher ?” that your propose is just plain incorrect and might confuse a native speaker hearing it.

Likewise, “J’ai rien foutu de la journée” (I didn’t do shit all day) can only be replaced by “J’ai rien fichu de la journée” and not by “J’ai rien fiché de la journée” (A correct sentence, but with a different meaning)

Alternative:

Qu’est-ce que j’en ai à :

• carrer ?
• branler ?
• péter ?

Oh wait, they are all as slang as ‘foutre’ 😐

Since ‘Qu’est-ce que j’en ai à… ?’ is a rhetorical question, you can replace it with affirmative: ‘J’en ai rien à…’.

We have a funny expression: ‘Je m’en tamponne l’oreille avec une babouche’.

A more polite version, with the same structure, is :

Qu’est-ce que j’en ai à faire ?

But it still is quite passive aggressive.

There are a lot of other idiomatics to convey that meaning, as said in the other answers.
I would recommend those common ones :

Je m’en fiche pas mal

Ça m’est bien égal

On s’en fout

A very vulgar expression is "on s’en bat les couilles" and its extension, which is commonly used as an equivalent "ça m’en touche une sans faire bouger l’autre".