Another thing is the following extract:

"Let's go back home. It's late." - "I'd rather stay out a little longer." - "I suppose we've got to go home ... time."

Is it "some" that should be here?

 Yes, because “some” refers to a more specific time and makes more sense with the context. The worry that “it’s late”, and “a little longer”, seems to suggest that a certain time has elapsed or is about to elapse.

 In this case “some” has a specific meaning, whereas “any” would not have.

 Please ask for further explanation, I’ll try my best…

Confused with pronouns

There are two sentences that I find very similar to each other in meaning but still there must be some trick if they give both of them. I'm supposed to insert some/any/no.

1. Will you have ... more tea? - Thank you.
2. Will you have ... more tea? You've had only one cup.

For the 1st one I think it's "some" as it's a usual offer of something to eat/drink.
For the 2nd one I guess it should be "any" but still don't understand why it can't be "some" as well.

Use "some" in positive statments, polite requests or an offer.

1. Would you like some more tea? - Yes please.. (an offer)

2. Sorry, we don't have any more tea! (negative response)

Use "any" in negative statments or in statments where it doesn't matter which of a range of Possibilities are available.

1. You can come over any day this week. (non-specific)

2. You can come over some day this week. (This statement leads to the question - Well… which day???)

3. Would you like some tea, you can choose any of the teas here on the menu?

 In answer to your question, you should have "some" as the answer to both of your sentences. Both sentences are an offer of something. (although, a bit more context would help!)

Another thing is the following extract:

"Let's go back home. It's late." - "I'd rather stay out a little longer." - "I suppose we've got to go home ... time."

Is it "some" that should be here?

Confused with pronouns

There are two sentences that I find very similar to each other in meaning but still there must be some trick if they give both of them. I'm supposed to insert some/any/no.

1. Will you have ... more tea? - Thank you.
2. Will you have ... more tea? You've had only one cup.

For the 1st one I think it's "some" as it's a usual offer of something to eat/drink.
For the 2nd one I guess it should be "any" but still don't understand why it can't be "some" as well.

One could use "any" for the second one, expressing the assumption that they have not drunk enough and should drink some more (one would highly stress accent "any").