Стоматолог сказала, что у меня чувствительная десна и усталая челюсть.

Because the dentist is female. In Russian, past-tense verbs agree with the speaker's or subject's gender:

• сказал = a man said
• сказала = a woman said

Even though стоматолог looks like a masculine noun in form, it can refer to either a male or a female dentist. The verb tells you the actual gender here.

Что means that and introduces the clause after said:

• Стоматолог сказала, что...
• The dentist said that...

In Russian, a comma is normally required before что when it introduces a subordinate clause like this.

Russian usually expresses possession with у + genitive rather than a direct verb like English have.

So:

• у меня literally means something like at me
• у меня чувствительная десна = I have a sensitive gum / my gum is sensitive

This is one of the most common Russian patterns, so it is worth getting used to early.

Because the preposition у requires the genitive case.

The pronoun changes like this:

• я = I
• меня = me / of me / at me, depending on the structure

So:

• у меня = at me / I have

You cannot use мне here, because мне is dative, not genitive.

Because in the у меня possession pattern, the thing possessed is usually in the nominative:

• у меня десна
• у меня челюсть

In this sentence, both nouns are the things being described as existing with me / in my case, so they stay in nominative singular.

Because adjectives must agree with the nouns they describe in gender, number, and case.

Here:

• десна is feminine singular nominative
• челюсть is feminine singular nominative

So the adjectives are also feminine singular nominative:

• чувствительная десна
• усталая челюсть

If the nouns were masculine, neuter, plural, or in another case, the adjective endings would change.

Yes. Nouns ending in -ь can be either masculine or feminine, so you have to learn their gender individually.

• челюсть is feminine
• that is why it takes усталая, not усталый

By contrast, десна is clearly feminine because it ends in -а.

Because Russian normally leaves out the present tense of to be.

English says:

• My gum is sensitive
• My jaw is tired

Russian just says:

• чувствительная десна
• усталая челюсть

with no separate word for is.

You may know есть, but that is not normally used here. Есть is more about existence or possession in a stronger sense, while this sentence is mainly describing qualities.

Russian can use the singular десна when talking about gum tissue or a particular gum area, even where English might prefer gums.

So the singular here is not strange in Russian. If someone wanted to speak about gums more generally or more explicitly in the plural, they could use дёсны.

Yes. Russian word order is fairly flexible.

This order is neutral and natural:

• Стоматолог сказала, что у меня чувствительная десна и усталая челюсть.

You could change the order, but that would usually shift the emphasis rather than the basic meaning. For example, putting усталая челюсть first would make the jaw stand out more.