Девочка Анны - what do you mean by that? "Девочка Анны" sounds strange, it means "Anna's girl".

If Анна is the girl's name, then you have to write "Девочка Анна" (the girl called Anna).

Утром, я увидел милую девочку в комнате подруги Ольга.

Утром я увидел милую девочку в комнате подруги Ольги.

1) Please remember that we do not ever use a comma after time modifiers (like "утром, вечером, вчера, сегодня, завтра, в прошлом году" etc.). It is a regular sentence member, there is no reason to separate it with a comma. I already told you that
2) In "подруга Ольга" the both nouns should decline: "в комнате подруги Ольги", "к подруге Ольге", "с подругой Ольгой" etc.

Я думаю, она сестра маленького мальчика, которого я увидел вчера вечером в парке.

1) Lexical note: "молодой мальчик" does not sound right. We use "молодой" about people of more or less adult, legal age. The actual range of "молодой" can be quite wide, depending on how you understand the youth, but usually it is somewhat between 14 and 30. The limits are not strict, of course.
However, "мальчик" means an underaged boy (definitely younger than 18 or even younger than 16). Usually, "мальчик" is from 0 (just born) up to 14-16. That is why I do not like "молодой мальчик". We would say "маленький мальчик". Yes, "маленький" means age when applied to children.

2) The conjunction "который" is mandatory and it cannot be just skipped as you did.

3) Although "которого я увидел вчера вечером в парке" is grammatically correct (there's no error in it!) I would suggest two changes:
- "которого вчера вечером я увидел в парке" is a better word order here;
- "которого вчера вечером я видел в парке" is even better. You can use perfective (увидел) but it implies that was very short (you were passing by, and you saw that young boy for a short moment), and it also implies you do not expect to see him again. That's why I suggest "видел" instead of "увидел".

BTW, in the first sentence (Утром я увидел милую девочку в комнате подруги Ольги) you can also use "видел" instead of "увидел", it would fit the context better.

Моя подруга сказал что она дочь её подруга Анна.

Моя подруга сказала, что она дочь её подруги Анны.

1) "Подруга" is she, not he. Подруга "сказала".

2) Please always use a comma to separate two clauses (which may be connected via "что", "чтобы", "который" etc.) I also noted it already before

3) "Она дочь её подруги Анны" (She is a daughter of her friend Ann). Here, Анна is the name of a friend of your girlfriend. In this case, the both nouns should be in genitive (дочь [кого? of whom?] - подруги Анны).
Another possibility: "Она дочь её подруги, Анна". (She is Ann, a daughter of her friend). In this case, Анна is the name of the girl. And she (Анна) is a daugther of her friend. It can be rephrased the following way to make it clearer: "Она дочь её подруги, Анна" = "Она Анна, дочь её подруги".

4) The whole sentence makes it difficult to understand due to duplicate use of "подруга", and due to overloading with pronouns (Моя подруга сказала, что она дочь её подруги Анны) - it is hard to understand even for a Russian. I would better rebuild the phrase:
Моя девушка (if she is really your girlfriend) сказала, что та маленькая девочка - дочь её подруги Анны.
or:
Моя девушка (if she is really your girlfriend) сказала, что та маленькая девочка - это Анна, дочь её подруги.
- depending on what you actually mean.

Но Анна уехала из дома.

No mistake here

But it turns out that Anna is the name of your girlfriend's friend actually. And that little girl is Anna's daughter.
So, my guess that the title of the story should be "девочка Анна" was wrong.

But you cannot say "девочка Анны". "Anna's girl" is very strange, as if the girl belongs to Anna. You should have titled the story as "Дочь Анны". Or, if you want to emphasize she is a little girl, you could title it "Маленькая девочка" (without specifying whose daughter she is). Or, if you want to be super-precise, title it "Маленькая девочка, дочь Анны"

Она ехала одного собранию на работу.

1) The "одного собранию" part does not make any sense to me. What did you try to say?

2) The rest of the sentence should be:
Она уехала на работу.

You used perfective in the previous sentence: "Но Анна уехала из дома". So you need in the next sentence, too. You just emphasize the fact "she has left home". And then you provide more details of where she left: "She has left for work".

3) Probably, you meant "she has left for work for a meeting"? Then, you should just add "на собрание": "Она уехала на работу на собрание".

Итак моя подруга заботится о ней милая, симпатичная младшая девочка.

Вот моя подруга и заботится о ней, о милой симпатичной маленькой девочке.

1) "итак" does not suit stylistically. This word (итак) is good for logical consequences when you explain something. It is more like "thus", "so we have", "it logically follows".

I understand you wanted to express "cause-result" chain. Ann has left home, so your girlfriend takes care of her daughter.
To express this, there is a colloquial construction "Вот ... и ...", the example is above.

2) First you used "... заботится о ней" and then you specified who is implied with "о ней". It is a bit complicated construction, but if you want to do so, you have to put the specifying part in the same case: "о ней, о маленькой девочке".

3) Again, "младший" does not fit here. "Младший" is a relative age (younger). If there were two girls, then you can call one of them "младшая девочка" and the other "старшая девочка".

Через десять минутов, я вышёл из комнаты улыбавшись.

Через десять минут я вышел из комнаты, улыбаясь.
1) одна минута, две (три, четыре) минуты, пять (6 .. 10) минут.

2) As I noted before, no comma is needed after time modifiers (через 10 минут).

3) The verb "вышел" is pronounced with the first syllable stressed: "вышел". There is an easy rule for that: the prefix "вы-" in a perfective verb always pulls the stress to itself. That is why "ё" is wrong here. Please compare (stressed vowels are bold):

шёл, пошёл, пришёл, ушёл, зашёл etc. but вышел,
летел, полетел, прилетел, улетел, залетел etc. but вылетел,
пил, допил, попил, запил etc. but выпил,
нёс, отнёс, унёс, понёс etc., but вынес,
...

4) We put a comma before a verbal adverb (деепричастие): "вышел, улыбаясь".

5) The most difficult thing: "улыбавшись" is theoretically possible. It is a past-tense imperfective деепричастие.
But according to the standard literary Russian, this form does not exist. There are only two kinds of verbal adverbs:
- imperfective (simultaneous actions);
- perfective (consecutive actions).
Your choice of the imperfective was absolutely right here. But in the standard literary Russian the деепричастие does not have tenses at all. You only need to express the action was simultaneous. Therefore, it is just enough to say "вышел, улыбаясь" (came out while smiling). But in colloquial sub-standard speech your version is possible.

Но где её брат, что я увидел в парке вчера вечером?

Но где её брат, что я увидел в парке вчера вечером?

Theoretically, correct. I would suggest some corrections, however:

Но где её брат, которого я видел в парке вчера вечером?

1) "который" is a better conjunction related to a noun than "что". You have a noun in the first clause (брат), moreover it is animate. Then you refer to it in the second clause using "что". It is not a mistake, but it does not sound good.

2) Why "видел" and not "увидел" is preferrable, - I have already explained it above.