Le, la and les are used to identify something that is known and unique.

e. g: le cahier.

Un, une and des are used to identify somthing that is not known and not unique.
e. g: J’ai une voiture.

Du, de la are used to identify a quantity.
e. g: il a mange’ du pain.

In the example above, it should be: Des haricots et du pain.

In fact, this is how it is written in the doulingo official site:
https://www.duolingo.com/comment/310405

Il y a un certain nombre de cas en français où l’article est omis:
L’article est habituellement absent devant :

- l' apposition : Sa mère, femme remarquable...
- l'attribut : Son fils est médecin.
- le nom en apostrophe : Compagnons, jurons de ne jamais nous rendre.
- certains compléments de noms : Une maison de bois.(et non du bois) Un esprit de synthèse, (et non un esprit de la synthèse) etc.
- devant les noms de jours et de mois s'ils ne sont pas accompagnés d'un élément subordonné : Nous nous verrons lundi.
- devant "minuit" et "midi" : Nous nous verrons à midi.
- devant des noms faisant partie d'une énumération dynamique : Livres, cahiers, stylos volaient à travers la classe.
- avec de nombreuses expressions figées : Crier victoire. Avoir peur. Perdre patience, etc.
- dans le style proverbial : Pierre qui roule n'amasse pas mousse.
- devant les noms qui se désignent eux-mêmes et dans les inscriptions : Axe vient du latin "axis" et signifie proprement "essieu. Voiture à vendre.

http://grammaire.cordial-enligne.fr/manuels/ART_OMIS.htm

TO answer the specific query: “So why can I say J’aime les fraises but not Vous vendez les chaussures?“

I see the distinction between les and des a bit differently from you:

les implies “all”, “the totality”, so «J’aime les fraises» means that you like all strawberries. The whole, all-encompasing category including their platonic ideal.

des implies a subset, some specific members of the category, so «J’aime des fraises» suggests that you like some particular ones on occasion.

Consequently Vous vendez les chaussures suggests that he/she is the seller of all the shoes.

Well, this is a tricky one for sure.

Let’s start at the beginning, there is 3 categories of article in French.

First come the definite articles:

• Singular male: le, l’, du, au
• Singular female: la, l’
• Plural: les, aux, des

You use them when what you’re talking about if define, both the speaker and the listener know the object definite. So it would be the equivalent of this/that in English. It is also used for the possessive form, when the belonging is obvious, like “My hand hurts” would be translated to “J’ai mal à la main”.

In the shoes example, in the case you want the red shinny shoes you found, you’ll ask the vendor “Do you sell the shoes?” then in French “Vendez-vous les chaussures?”. Because you’re talking about those particular shoes.

Second is the undefined articles:

• Singular male: un
• Singular female: une
• Plural: des

You use them when what you’re talking about is undefined, or can’t be defined. It can also be used for emphasis.

Still with the shoes, if you want to ask if the store sale any shoe you’ll say “Vendez-vous des chaussures?”.

For emphasis I’ll give you an example “He is so stupid” could be said “Il est d’un bête”

The last one is the partitive article:

• du, de la, des

It used for what can not be quantified, like bread. Basically it’s some in English.

The Strawberries singularity

So, what’s up with the strawberries ? It’s cool to have definite all the articles, but it might not help us to get why is happening in your example.

“— Hey, what are you doing?

— I’m eating strawberries”

In this case, you’re eating any strawberries, then you will use “des”.

“— Hey, what are you doing?

— I’m eating the last strawberries”

In this case, you’re not eating any strawberries, you’re eating the last one, damn you! So you’ll use “les”.

“— Hey, what’s you favorite fruit?

— It’s strawberries”

Here comes the realm of liking, and yes you were right, liking is kind of a particular case somehow but not really if you think about it. You’re asking what kind of fruit you love, and thus it is a defined kind. Therefore you’ll use “les”, because you’re referring to all the strawberries, you love them all as a unique kind of fruits, not some undefined part of them.

The des grey area

You must have noticed it. Why des is everywhere ?

The reason is simple, for the first category it is des is the contraction of de les. For example “She is the kids’s nanny” would be said “Elle est la nourrice des enfants”. For the second category, if it really des. And for the third category, des is actually used for some particular words like for bread you’ll use du and for spinach you’ll use des.

I hope I helped you to understand this French nonsense.
Tell me is there is some part you don’t get so I can explain more.