Learn English – “I should be doing the homework.” Vs. “I should do the homework.”

1. Who sent those flowers?- I'm not sure. It could have been your mother.

The above is correct written form. There's a lot of latitude with speaking, of course. Here are some alternatives:

• Who sent these flowers? Couda' been your mother.
• Who sent these flowers? Maybe your mother?
• Who sent these flowers? Could be your mother. <- not likely but understandable. This and other types of "incorrect and slightly odd" spoken responses happens often enough. It can make one think a second longer to fill in the gap or an opportunity to make a joke: "Oh! My mother turned into flowers? So who do you think sent my mother? (haha)"

2) He could have been Prime Minister now if he hadn't decided to leave politics.

The above is correct written form. You could find a place for your "could be" form in spoken language as a parallel, emphatic, emotional voice. The following illustrates this with a similar construction, but with some necessary context:

• Man #1: "It's not so bad...I'm doing well now... I've worked my way up to vice president of the Acme Corp!"
• Man's Tough Friend: "Don't give me that! You could be CEO now if you hadn't messed everything up in the first place."

3) We could have spent today at the beach, but we thought it was going to rain, so we decided not to.

The above third case is not correctly worded as you stated above. It doesn't make sense to say "I could have done X, but since Y I decided not to." That's because the decision was totally volitional. The format is "I would have done X, but since Y I decided not to." So the original sentence above would be more natural as:

• "We [were going to spend | would have spent] [the day | today] at the beach, but we thought it was going to rain, so we decided not to."

To use your original sentence wording ("could have spent"), it would need to be something less within the speaker's control, and perhaps an expression of disappointment like this:

• I could have spent the day at the beach, but Wendy got sick so I had to stay and babysit. Hrmph! (Note that this is also volitional in an absolute sense; yet has a sense of blame.)

To make it your alternative "could be spending" suggests a greater disappointment that is still ongoing and not fully accepted:

• On the phone: "I could be spending the day at the beach, right now, with all my friends, but Wendy just had to get sick and so I'm just stuck here baby sitting. Hrmph! (Note that this is also volitional in an absolute sense; but has a greater sense of disappointment and blame.)

The most mature way to handle the situation would be decisive and volitional:

• "Yes, I know... I was planning on spending the day at the beach with you guys. But Wendy got sick so I'm staying home to take care of her today. I'm certainly not going out if Wendy needs me. Have fun... I gotta go now. Bye."

These are all meaningful, but the meaning is changed and the context in which they could be used would be changed. "Would" is more often used to discuss expectations (that one feels certain about) in projected courses of action, rather than hopes or expectations in the current situation. You wouldn't use "would" there except in limited circumstances, when you are very confident in your predictions. I think a lot of this falls out of the use of "should" to describe model behavior or what is 'right' (like def. 3 here). Incidentally, I think (without proof) that that is why we don't use "should" to describe expectations that go awry, like the train being late: we certainly don't think that our hopes or expectations ought to be denied!

1. with "should" is a prediction; with "would" the speaker is certain enough to, for instance, discount the possibility of it not having been received. ("They would have it by now, so that can't be the reason they haven't responded...")

2. "should" states a probable expectation, while "would" is more like a certainty in a hypothetical plan ("We would get there by six, which would give us time to change before dinner...")

3. is tough. "We wouldn't have to wait much longer" is an acceptable utterance with an appropriate context: if I'm making a case for my plan (staying in line) over your plan (leaving the line). The implied full thought is "we wouldn't have to wait much longer [if we agreed to stay]." Here we are discussing the expected-to-be-certain details of possible plans, rather than stating our hopes about the future.

4. would work if you'd said "there were no reports of delays, the train would be on time [and that is why I am worried that your sister hasn't arrived yet]". This is related to sentence 1; I am stating a prediction about which I feel certain enough that I discount the train being late as a possible reason that your sister is delayed, and start worrying that something else could have happened to her.

5. Meaning is completely changed--we don't use "should" to emphasize things that happen contrary to expectation. Sometimes "The train would be late!" (with emphasis) is used expressing frustration that the train is late. The resulting statement is whining. A more full example: "The train would be late on the day I have an interview! This always happens to me!"

So: the distinction as I see it is pretty much what your edited addition says. "Should" suggests a more tentative attitude than "would," which may be from genuine uncertainty, or from an attempt to be more polite or more emotionally removed.

Both "should" and "would" are used to discuss an event about which the speaker is not completely certain or confident. But "would" seems to have the implication of talking about the details of a hypothetical world, future course of action, proposed plan, etc. "Should" is used to make less confident predictions about the future.