Je pose ma brosse à dents sur le lavabo.

Here, pose comes from the French verb poser, which often means to put down, to place, or to set something somewhere.

It is related to English pose historically, but in this sentence it does not mean striking a position for a photo. It simply means placing an object down:

• Je pose mon livre sur la table. = I put my book on the table.
• Je pose ma brosse à dents sur le lavabo. = I put my toothbrush on the sink.

Both can work, but they are not exactly the same.

• mettre = a very general to put
• poser = to place / to set down, often onto a surface

So poser sounds a bit more specific here because the toothbrush is being placed on the sink.

Compare:

• Je mets ma brosse à dents sur le lavabo. = I put my toothbrush on the sink.
• Je pose ma brosse à dents sur le lavabo. = I set my toothbrush down on the sink.

Both are natural, but poser matches the idea of placing something onto a surface especially well.

Because brosse is a feminine noun in French.

French possessive adjectives must agree with the noun being possessed, not with the owner. So:

• ma = my + feminine singular noun
• mon = my + masculine singular noun
• mes = my + plural noun

Since une brosse is feminine, you say:

• ma brosse

That is why you get ma brosse à dents.

In brosse à dents, the à shows purpose or function. Literally, it is something like brush for teeth.

This is very common in French compound nouns:

• une brosse à cheveux = a hairbrush
• une tasse à café = a coffee cup
• une machine à laver = a washing machine

So une brosse à dents is the normal French way to say toothbrush.

Because a toothbrush is used for teeth, not just one tooth.

So French says:

• une brosse à dents = a toothbrush
  literally, a brush for teeth

This is similar to English, where toothbrush is singular as a whole object, but it is for brushing all your teeth.

In French, you normally need an article before a noun. So lavabo usually appears with something like:

• le lavabo = the sink
• un lavabo = a sink

French does not usually omit the article the way English sometimes can.

So:

• sur le lavabo = on the sink

is the natural form.

Le lavabo refers to a specific, identifiable sink, usually the one in the room or the one both speaker and listener understand from context.

So:

• sur le lavabo = on the sink

If you said sur un lavabo, it would sound more like on a sink, with less specificity.

French often uses the definite article when the object is understood from the situation.

Not exactly.

Le lavabo usually means a bathroom sink or washbasin.

If you mean a kitchen sink, French more often uses:

• l’évier

So:

• sur le lavabo suggests a bathroom setting
• sur l’évier would suggest a kitchen setting

Sur means on or on top of.

It is used because the toothbrush is being placed on the surface of the sink.

Compare:

• sur le lavabo = on the sink
• dans le lavabo = in the sink

So if the toothbrush is resting on the edge or top area, sur makes sense. If it were lying inside the basin itself, dans might be more appropriate.

It is the present tense: more specifically, the present indicative of poser.

• je pose = I place / I put / I am placing / I am putting

French present tense can cover several ideas that English often separates:

• I put my toothbrush on the sink
• I am putting my toothbrush on the sink

The exact meaning depends on context.

Yes. Poser has several common meanings in French, depending on the object that follows.

For example:

• poser un livre = to put down a book
• poser une question = to ask a question
• poser un problème = to cause a problem / to present a problem

So the verb is very flexible. In your sentence, because the object is ma brosse à dents, the meaning is clearly the physical one: to place / put down.

No, it is very standard French word order:

• Je = subject
• pose = verb
• ma brosse à dents = direct object
• sur le lavabo = prepositional phrase showing location

So the pattern is:

Subject + Verb + Object + Place

That is one of the most common sentence patterns in French.