Going crazy with particles! (and +)

• doru wo motte imasu ka? (do you have some dollars?)
• sushi wa taberaremasu ka? (can you eat sushi?)
  
• biiru ga kaemasu (I can buy beer)

kurapan said:

Let me try to help you shed some light on it.
Let's start with the topic. What is a topic? Topic is the main theme of the sentence, the thing that the sentence or a series of sentences is about. To illustrate this, let me show a simple example.

Watashi wa John desu. 22-sai desu. Amerika kara kimashita. Ongaku to eiga ga suki desu. Kasei ni iku no ga yume desu.
(I'm John. I'm 22 years old. I come from America. I like music and movies. Going to Mars is my dream.)

Here the topic is watashi, that is 'I'. The speaker talks about himself over the span of several sentences. You could insert it in each of the above sentences, but it's not necessary and no Japanese would do it because the meaning is totally clear even without it. How come? Because we stated the topic of the speech in the first sentence and it remains as the topic until you introduce a new topic. If I tried to illustrate it in English, it would something like:
I'm John. 22 years old. Come from America. Like music and movies. Going to Mars is dream.

Let's have a look at the last sentence. The subject of the sentence is iku no, meaning 'going'. The main topic is 'I', you're talking about yourself, doing a self-introduction, and your dream is one of the things you talk about but it's not the most important thing that you want to convey, it's just a part of the list.
Now, let's have a look at a slightly different series of sentences.
I'm John. I'm 22 years old. I come from America. I like music and movies. My dream is to go to Mars.
The change in the last sentence might not seem as a huge difference. But from grammatical point of view it turns from an object into a subject and now it sounds as a more important thing.
Now let's have a look at a slightly different sentence in Japanese.

Watashi wa John desu. 22-sai desu. Amerika kara kimashita. Ongaku to eiga ga suki desu. Watashi no yume wa kasei ni iku koto desu.

The last sentence introduces a new topic, 'dream'. When we rephrase the introduction like this, making 'dream' a topic, the dream becomes an important point that we want to convey. The topic of the speech changes from 'I' to '(my) dream'.

Another example. Let's say you're talking about a picture of several cars and want to say which one you like. You could say something like:
Watashi wa kono kuruma da.
Meaning, 'I (like) this car'. But if you said:
Watashi ga kono kuruma da
it would sound as if you were saying, 'I am this car', because it grammatically links 'I' and the existential verb.

Another difference between 'wa' and 'ga' is very apparent in questions.
'What is this car?' Kono kuruma wa nan desu ka?
'It's Benz' Kono kuruma wa Bentsu desu

'Which car is Benz?' Dono kuruma ga Bentsu desu ka? (alt.: Bentsu wa docchi desu ka?)
'This car is Benz' Kono kuruma ga Bentsu desu.

Another purpose of topic is introducing contrast.
Okanemochi to kekkon suru no wa yume desu. (Marrying a rich person is my dream.)
Watashi wa kasei ni iku no ga yume desu. (My dream is to travel to Mars.)

A trick that doesn't always work is trying to omit the word before 'wa/ga'.
context: you just came home and someone asked you what you did
(Watashi wa/ga) kaimono ni ikimashita.
'(I) went shopping'
Is the information we want to convey clear even without the word? Yes, it is. Why? Well, it's the topic here. The topic here is obvious from the context (which is not always true of course) and most importantly, the topic is not so strongly linked to the main point of the sentence. Therefore, you write wa.

Purin wo tabeta no wa dare desu ka? (Who ate the pudding?)
(Watashi wa/ga) purin wo tabemashita.
'(I) ate the pudding.'
Someone asks who ate the pudding. If you only say 'ate the pudding', the desired information won't be conveyed. In such a case where the sentence becomes just meaningless bullshit without the word before the particle, you should probably use ga there.

Lastly, let's have a look at your three sentences.
Doru wo motteimasu ka?
This sentence for some reasons sounds weird to me without any context. But anyway, it's just a simple question whether you have dollars. In some context, you could use Doru wa motteimasu ka? to introduce contrast, for example if you asked about Chinese yuan first. Or you could just simply use it as a topic of the speech.

Sushi wa taberaremasu ka?
Sushi is topic here as the one who asks the question wants to know if you can eat sushi.

Biiru ga kaemasu
Again, hard to talk about this one without context. Let's make some then.
'What can you buy there?'
Biiru ga kaemasu. (You can buy beer.)
Heisha no shokudou ha sugoi desu yo. Biiru ga kaemasu. (Our company's canteen is awesome. You can buy beer there.)
If there was wa in the sentence, you could say
A: Heisha no shokudou ha biiru to haibooru ga kaemasu. (You can buy beer and highballs in our company's cafeteria.)
B: Sugoi! Heisha no shokudou de wa haibooru ga kaenai desu. Biiru ha kaemasu. (Wow! You can't buy highballs in our company's cafeteria. You can buy beer, though.)

---

Watashi no tokoro isn't wrong, but I have only heard it in limited context unlike more frequent watashi no ie, watashinchi, or simply uchi. (Feel free to replace watashi with any pronoun).
The context I remember having heard watashi no tokoro the most is when you suggest 'going to your place'. Most likely to netflix and chill.
Watashi no tokoro ni is also used to say 'to me'. As in 'I hope happiness comes to me too'. So it's more general and doesn't only mean your house but also your surroundings, your life.

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Toritoribe said:

An explanation from another point of view;
Ga can be used to indicate the direct object of the potential form of verbs, the -tai form, state verbs such like wakaru, iru or dekiru, adjectives that express feelings/emotions/sense (suki da, kirai da, nikui, kowai, itai, etc.). Thus, sushi ga taberaremasu ka? or biiru wo kaemasu is also valid, but doru ga motte imasu ka? is invalid since motte imasu is not the potential form.

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Toritoribe said:

An explanation from another point of view;
Ga can be used to indicate the direct object of the potential form of verbs, the -tai form, state verbs such like wakaru, iru or dekiru, adjectives that express feelings/emotions/sense (suki da, kirai da, nikui, kowai, itai, etc.). Thus, sushi ga taberaremasu ka? or biiru wo kaemasu is also valid, but doru ga motte imasu ka? is invalid since motte imasu is not the potential form.

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が の意味
［格助］名詞または名詞に準じる語に付く。
２ 希望・好悪・能力などの対象を示す。「水が飲みたい」「紅茶が好きだ」「中国語が話せる」

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が
一 （ 格助 ）
体言および体言に相当するものに付く。
２希望・能力・好悪などの対象になるものを表す。 「リンゴ－たべたい」 「あの人－好きだ」

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