Yes, the exercise wants you to write the first conditional as a second conditional. I've never heard it called "oblique mood". The meaning does change in second conditional: you are less sure or even simply imagining an unlikely situation. The first conditional tells me that you think the possibility is likely. "Should" actually is part of a proper second conditional sentence. In this case, you understand "should" as the past form of "shall" (the same way that "would" is the past form of "will"). This use of "should" means an indirect obligation. It's often replaced my "would" in modern days, but you still really need to know how "shall/should" work. In many cases they are not interchangeable.

If it were cold, we would put on our warm coats. If my friends came to see me, I would be very glad. Both of these sentences are Hypothetical. You are not saying that it was cold, but what you would have done, if it was cold at some point in the past. In normal usage, Should is usually used to recommend future actions. So in your sentence above you are saying "If it were cold (at any point in the past), then I should (in the future, based on a condition) put on my coat" You could say 'should have' when referring to things in the past, but only when contemplating things that did happen, and what you would do differently if the same thing happened again. So you could say 'when it was cold, I should have put on my coat' (but i did not) Having said all of that, some speakers in England would 'Should' the way you have, but to most people it would sound stiff, and it would imply that the listener should also be glad if their friends had come. This might be considered condescending by many. "Are "I should be happy" and "I would be happy" interchangeable" These are not really interchangeable. The first is comptemplative - "I should be happy (but I am not)". The second is conditional. "I would be happy (if something occured)"

a) If it is cold, we'll put on our warm coats. If it were cold, we should put on our warm coats. b) If my friends come to see me, I'll be very glad. If my friends came to see me, I should be very glad. This looks like out-dated English. You'll see it in older books. I don't think you'll see it in modern day speech though.

I should be happy and I would be happy aren't interchangeable. I should be happy/we should put on our coats- is more of an obligation. At least, in America we wouldn't really say "If it were cold, we should put on our warm coats" To me it sounds odd. If I were saying this phrase, I would say "If it were cold, we would put on our warm coats." Because of the "if" and then use of the past tense, you should use "would." Would is conditional/hypothetical. "Should" could be used if you use "if" and then present tense. Ex. "If it rains, we should put on our warm coats"

Time to get a new book I think! As the others have said, nowadays, 'should' implies obligation. A native speaker would say: If it were cold, we would put our coats on.