I see a mix of informations there. Let’s make it clear:

• An vs année: meaning the same idea but are not used in the same way. An is a count, I’m 20yo: j’ai 20 ans; Saying j’ai 20 années, is not grammatically wrong but not used in that sense. But Depuis 20 années, For 20 years, indicates a duration, the time that has passed. An refers to a date even this is not explicitly indicated.

• Jour and journée: Jour is a day of 24 hours. A journée, is not used for a duration because the meaning is not the same, a journée is the period of the day where the sun is up. It will be followed by the night, and starts with the morning (matinée).

So for your sentence: Vous enseignez ici depuis plusieurs années / jours / mois is the correct way to express the duration and the time that has passed. Using journées and ans doesn’t match unless you add a count: Vous enseignez depuis 5 ans.

The translations are correct, but an vs. année doesn’t work quite like jour vs. journée.

"Plusieurs ans" is extremely rare, and would probably be considered incorrect in most cases. As seen here:

Quant au mot année, il indique plutôt l’« approximation d’une durée »

So when using des, plusieurs, tant d’, quelques, etc. you’d have to use années. When a precise number is given, an is generally used, but année can work too. It’s a little more literary, and it tends to focus on time passed a little more. And it doesn’t work in phrases like "dans 5 ans" or "il y a 5 ans".

The choice can also be purely stylistic or based on syntax and how the sentence sounds, rather than meaning. For example, adding an adjective restricts what you can use:

Trois ans (factual, neutral)

Trois années (more literary or to focus on duration)

Trois longs ans (incorrect)

Trois longues années

I No, those translations are not correct.
The translation of "Vous enseignez ici depuis plusieurs années." is this.

• You’ve taught here for several years.

The explanation of this translation follows from the fact that the present in this case is used to talk about something that started in the past, goes on in the present and (most of the time) is expected to go on in the future.
From this English grammar link we get this.

• We use the present perfect: • for something that started in the past and continues in the present:
  They’ve been married for nearly fifty years.

From the same link we get this.

• We normally use the present perfect continuous to emphasise that something is still continuing in the present:
  It’s been raining for hours.

Therefore, another possible translation is as follows.

• You’ve been teaching here for several years.

II the use of "jours" is an error, a strange error by the way.

III In this context you can only use "année".

If you want to emphasize that the number of years is great then you can use an expression in which "an" is correct.

• Vous enseignez ici depuis des ans et des ans.
• Ça fait des ans et des ans que vous enseignez ici.

Nevertheless, "des années et des années" is used more often (ngram) for the same purpose.

• Vous enseignez ici depuis des années et des années.
• Ça fait des années et des années que vous enseignez ici.

I once answered a similar question:

The adding of "ée" to a noun generally indicate the contents (concrete or abstract) of the original noun: For example, une "assiettée" means the contents of a plate, meanwhile "assiette" is the plate itself. Une "fournée" means the contents of the oven ("le four"), une "charretée" means a cartload, contents of the cart ("la charrette"). On the abstract point, the principle is the same: the "matinée" is the duration of the morning ("matin"), meanwhile "matin" is the position in the order of the day, but not a duration. the same for an/année, jour/journée and so on. But it is true that through the centuries, the difference between "matin" and "matinée", for example, is not consciously established for most people, so they often use one for the other.