Meidän täytyy selvittää ongelma.

Because this sentence uses a common Finnish necessity construction:

• minun täytyy = I have to
• sinun täytyy = you have to
• meidän täytyy = we have to

So meidän is the genitive form of me.
A native English speaker may recognize meidän as also meaning our, and that is true in other contexts, but here it is not a possessive. Here it marks who has the obligation.

It means we must / we have to + verb.

So the full pattern is:

• [genitive form of person] + täytyy + infinitive

Examples:

• Minun täytyy mennä. = I have to go.
• Hänen täytyy odottaa. = He/she has to wait.
• Meidän täytyy selvittää ongelma. = We have to sort out the problem.

This is one of the most common ways to express necessity in Finnish.

Because täytyy is a special verb in this use. It does not normally take personal endings the way many other Finnish verbs do.

So Finnish does not say something like we-must with a special we ending on the verb. Instead, the person is shown by the genitive word:

• minun täytyy
• sinun täytyy
• meidän täytyy

So the idea of who must do it is shown by meidän, not by changing täytyy.

Because after täytyy, the next verb is usually in the first infinitive, which is the basic dictionary form of the verb.

So:

• täytyy mennä = have to go
• täytyy tehdä = have to do
• täytyy selvittää = have to sort out / clarify

English uses to before the infinitive, but Finnish does not use a separate word like to here. The infinitive ending is built into the verb form itself.

Selvittää can mean things like:

• to clarify
• to sort out
• to figure out
• to investigate
• to resolve

In Meidän täytyy selvittää ongelma, a natural English translation is often:

• We have to sort out the problem
• We need to figure out the problem
• We must resolve the problem

The exact English choice depends on context.

Yes. This is a very common and important distinction.

• selvittää is usually transitive: it takes an object
  
  • selvittää ongelma = sort out / clarify the problem
• selvitä is usually intransitive: it does not take a direct object in the same way
  
  • selvitä ongelmasta = cope with / get through the problem
  • selvitä hengissä = survive

So in this sentence, selvittää is used because ongelma is the thing being dealt with directly.

This is a very good question, because learners often expect ongelman.

In many ordinary Finnish sentences, a complete singular object is in the genitive:

• Me selvitämme ongelman. = We will sort out the problem.

But in constructions like täytyy + infinitive, the object is often nominative singular instead:

• Meidän täytyy selvittää ongelma.

So ongelma here is normal.

A useful comparison:

• Me selvitämme ongelman.
• Meidän täytyy selvittää ongelma.

Same basic object idea, but the case changes because the sentence structure is different.

Yes, in some contexts the partitive can appear, but it changes the feel of the sentence.

Very roughly:

• ongelma = the problem as a whole, something to be fully dealt with
• ongelmaa = the problem in a more incomplete, ongoing, or partial sense

In this sentence, ongelma is the natural choice because the meaning is that the problem needs to be dealt with as a complete task.

Yes, Finnish word order is fairly flexible, but the neutral order here is:

• Meidän täytyy selvittää ongelma.

That order is the most natural if you are simply stating the idea.

Other word orders are possible for emphasis or contrast, for example if you especially want to focus on the problem or on us, but they are less neutral. For a learner, the given order is the best default pattern to remember.

Yes. A few common alternatives are:

• Meidän pitää selvittää ongelma.
• Meidän on selvitettävä ongelma.

These are all close in meaning to Meidän täytyy selvittää ongelma.

Very roughly:

• täytyy = very common, standard
• pitää = also very common, often feels a bit more everyday
• on selvitettävä = more formal or more forceful

For most everyday learning purposes, meidän täytyy... is an excellent pattern to know.