2 normal yet critical questions

[Siang]

They are quite thought-provoking questions.
Question 1:
1. I don't want to go out alone. Will you come with me? (intention)
2. I don't want to go out alone. Do you come with me?

In the sentence 1, the speaker ask with an intention. But, what about the sentence 2? Is its meaning close to the sentence 1?

Question 2:
I lost my key in the morning. I have found it now.

when I combine them into one sentence with a defining-clause, I get this answer.
I have found the key that I lost. (without time adjunct)

When I add the time adjuncts into the sentence, I get this.
I have now found the key that I lost in the morning.
OR
I have found the key now that I lost in the morning.

Do you find any ambiguities in the 2 sentence above? or They seem all right.

 

[Tdol]

2. I don't want to go out alone. Do you come with me?

This doesn't work for me- you could have 'are you coming'.

 

[Raymott]

When I add the time adjuncts into the sentence, I get this.
I have now found the key that I lost in the morning.
OR
I have found the key now that I lost in the morning.

Do you find any ambiguities in the 2 sentence above? or They seem all right.

They are both alright. You don't need 'now', and I'd use 'this morning' if you mean the morning of today.
I have found the key that I lost this morning.
There is no possibility of it meaning that you found the key in the morning. That is, it doesn't mean, "This morning, I found the key that I lost."
The main reason it can't mean this is because you can't use the present perfect "I've found" with a specific time like "in the morning" or "this morning". So, "this morning" has to refer to 'lost'.

There is ambiguity in this sentence:
I found the key that I lost this morning.