Je mets mon cartable près de la porte.

Because mets is the first-person singular form of the verb mettre in the present tense.

• je mets = I put
• tu mets = you put
• il/elle/on met = he/she/one puts

French verbs change form depending on the subject, so je needs mets.

It comes from the verb mettre, which often means to put, to place, or sometimes to set depending on context.

In this sentence, mettre means to put/place something somewhere:

• Je mets mon cartable près de la porte.
• I put my schoolbag near the door.

Because cartable is a masculine singular noun, so the correct possessive adjective is mon.

French possessives agree with the noun owned, not with the owner.

So:

• mon cartable = my schoolbag
• ma porte = my door
• mes livres = my books

Even though I am the speaker, the form depends on cartable, not on the speaker.

Cartable usually means a schoolbag, satchel, or a bag used for carrying school things. It is not the most general word for any kind of bag.

For example:

• un sac = a bag
• un cartable = a schoolbag / satchel

So cartable has a more specific feel than just bag.

Près de is a fixed French expression meaning near or close to.

You should learn it as a whole unit:

• près de la porte = near the door
• près de l’école = near the school
• près du mur = near the wall

The de is required after près. You cannot just say près la porte.

Because porte is a feminine singular noun:

• le is masculine
• la is feminine

Since près de is followed by la porte, you get:

• près de la porte

French only contracts de + le and de + les:

• de + le = du
• de + les = des

But:

• de + la stays de la
• de + l’ stays de l’

So:

• près du bureau = near the desk
• près de la porte = near the door

Because in French, de contracts only with le and les:

• de + le → du
• de + les → des

But there is no contraction with la:

• de + la → de la

So près de la porte is correct, because porte is feminine.

Mostly yes. The sentence follows a very familiar pattern:

• Je = subject
• mets = verb
• mon cartable = direct object
• près de la porte = place expression

So it is basically:

I + put + my schoolbag + near the door

That makes this sentence fairly straightforward for an English speaker.

Yes. Poser is also very natural here.

• mettre = to put / place
• poser = to set down / put down

In many everyday situations, both work:

• Je mets mon cartable près de la porte.
• Je pose mon cartable près de la porte.

Poser can sound a little more like physically setting something down on a surface or in a spot, while mettre is more general.

Je mets is pronounced approximately like zhuh may.

A few useful points:

• The j in je sounds like the s in measure
• The e in je is a soft, reduced vowel
• In mets, the final s is silent
• mets, met, and mais can sound very similar or identical in normal speech

So although mets is spelled with -ts, you do not pronounce the final s.

No, but it is one of the most common and useful ways.

Other possibilities include:

• proche de = close to
• à côté de = next to / beside

The differences matter:

• près de la porte = near the door
• à côté de la porte = next to the door

So près de means general closeness, not necessarily directly beside something.