Learn English – Couldn’t have happened

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."

As Mooseman says, "could" indicates that something is possible. "I could help you" means "I am capable of helping you". Exactly what the speaker means by this can be ambiguous. He might mean, "Yes, I am indeed capable of helping you and so I will." Or he might mean, "I am capable of helping you, but I am not going to help you because what you ask is too difficult or boring or I don't like you" etc. You would have to hear or read the larger context to know which is meant. In speech, you often rely on tone of voice. Like if someone says, "Well, I COULD help you", putting emphasis on the word "could", that likely means that he won't. But if he says "Yes, I could help you" in a normal tone, that probably means that he will help.