I Only for a few well-known verbs is it possible to use a "partial" negation.

(Français Facile)
1 dans certains proverbes et expressions
♦ Qui ne dit mot consent. ♦ Qu’à cela ne tienne.
2 avec les verbes avoir, savoir et pouvoir suivi de que interrogatif et d’un infinitif
♦ Il ne sait que dire. ♦ Il ne sait pas quoi dire.
3 avec les verbes cesser, pouvoir et savoir

As concerns the verbal locution "avoir envie", there is no way to omit "pas". So, the correct wording is as follows (which does not say that the whole assemblage should be idiomatic).

• Je n’ai pas envie de course à pied aujourd’hui.

II Both a verb in the infinitive and a noun are possible complements for the verbal expression "avoir envie", which is constructed in both cases with the preposition "de". However, the construction with a verb is common, whereas that with a noun is not found as much. Here is a selection of the usages concerning the use of "envie" in the verbal locution "avoir envie" ; it is found in the TLFi (user LPH’s bold type).

A. [Souvent sans art. dans des loc. verbales] Besoin, désir plus ou moins violent.
1. [En parlant d’un besoin physiol.] Avoir envie de boire, de dormir, de manger
2. [En parlant d’un besoin phys., d’une décharge nerveuse qui voudrait se manifester] Avoir envie de marcher, de pleurer, de rire
3. [En parlant d’un désir plus général relevant de l’affectivité, du rêve, du besoin d’action ou de possession, de l’ambition, de toutes les pulsions psychiques]
♦ Avoir envie de voir, de revoir qqn.
SYNT. avoir envie de se marier, de mourir, de parler, de partir
a) Avoir envie de qqc.
b) Avoir envie de qqn. Éprouver le désir sexuel pour cette personne.
♦ Et il la regarde parce qu’il a envie d’elle. Absol.

Usage is variegated, and it is rather hard to say sometimes whether a form is idiomatic.
In the case of a verb to which corresponds the noun of an activity, the verb in the infinitive is used, not the noun (courir/course à pied, peindre/peinture, chanter/chant, danser/danse,…) ; in order to use the noun (this usage is rarer) another verb is necessary, often this verb is "faire" ; sometimes no verb seems natural, and only complicated construction depending on the versatility of the speaker/writer; sometimes there is no verb.

• envie de courir/envie de faire de la course à pied
• envie de peindre/envie de faire de la peinture
• envie de chanter/envie de faire du chant (not natural)/[envie de se consacrer à ses leçons de chant/envie de travailler son chant (in the case singing is taken up as a profession)]
• avoir envie de nager/avoir envie de faire de la natation
• (no verb)/avoir envie de faire de la compétition
• (no verb)/avoir envie d’exercer un métier

When there corresponds a noun to the verb, this noun standing with respect to this verb in a relation of grammatical object, then again, the verb is preferred, but not always.

• envie de manger (used absolutely)/envie de nourriture (ref.)
• avoir envie de fraises/envie de manger des fraises (ref.)

In some cases both are equally likely and the added verb option is also available.

• Il a envie de boire un verre./Il a envie d’un verre.
• Nous avons envie de nous doucher./Nous avons envie d’une douche./Nous avons envie de prendre une douche.
• J’ai envie de voir un film. (only form possible)/J’ai envie d’un film. (not said at all)

In some cases "avoir envie" will not be idiomatic, and instead "avoir besoin" will be correct.

• j’ai envie d’un somme./J’ai besoin d’un somme.

Nevertheless, a partial rule can be stated. If the uncountable noun is not usually an option, the countable one obtained by reclassification from the countable one is sometimes proper for using with the expression; at times the noun has to be qualified.

• avait envie de bière/avait envie d’une bière (ref.; The use of the uncountable noun is rare, will not be idiomatic in usual daily talk, but in a certain literary context can be quite correct.)
• avoir envie de pain/avoir envie d’un petit pain (This is not just reclassification from uncountable to countable, the sort of bread is also changed.)
• avait envie de café/avait envie d’un café (ref.; notice the great frequency for the countable noun)

In the case of "course à pied" you do not find "envie de course"; therefore, you are even less likely to find "envie de course à pied"; however you do find a lot "faire de la course à pied"; this is shown in this ngram. This means that you are not likely to hear or read "Jai envie de course." or "Jai envie de course à pied."; instead, people will say "J’ai envie de faire de la course à pied.".

In conclusion, let’s say that this type of usage is sometimes possible (not in the case of "course à pied"), but there is no rule that permits to prescribe all the cases.
Duolingo appears then to be in the right.

"‘Je n’ai envie pas de courir" is wrong, period. The word pas must go immediately after the verb.

"‘Je n’ai pas envie de courir" is correct and idiomatic.

"Je n’ai pas envie de course à pied" is grammatically correct and comprehensible, but it is not something a French speaker would normally say. You can say "avoir envie de" + noun for something that you want to get. For something that you want to do, the construction is "avoir envie de" + verb (infinitive). The distinction is sometimes subtle but for a given thing there’s usually only one idiomatic choice. For something that you wish would happen, there’s "avoir envie que …".

J’ai envie de courir. (I want to run.)

J’ai envie de manger du gâteau. (I want to eat cake.)

J’ai envie de gâteau. (I want some cake.)

J’ai envie de vacances. (I want a vacation. The vacation is considered something you get.)

J’ai envie de nature. (I want to experience some nature.)

J’ai envie de toi. (I want you, i.e. I want to have sex with you.)

J’ai envie que ça s’arrête. (I want it to stop.)

(I translated "envie" by "want" everywhere for simplicity, but it’s actually weaker, often closer to "would like".)

Note the spelling "course à pied". Accents are not optional. Some native speakers would get it wrong or not bother going the accent on a medium where it’s inconvenient though.