Yes, you can leave the green on. You can scrape the carrots or just wash them, either is fine. If I were leaving the tops on I'd probably just wash them for the 'whole carrot' look. Boil them until hot and soft but not mushy, you still want them to have a bit of bite. Check after five, but it might take 7 to 10 depending on the carrots.
You can boil asparagus, but it's quite a delicate vegetable. I'd probably steam them.
Shallots - you can, or you can chop them the way you would an onion.
You can also chop the carrot tops and use them as a garnish as you would, say, parsley.
The tips of asparagus need less cooking time than the stems. If you don't have an asparagus pan and who does trim the stems to the same length and loosely tie the stalks in a bundle with kitchen string. Put them in a deep saucepan and pour in boiling water without covering the tips. Cover the pan with a tight-fitting lid and cook for about 15 minutes, so the stems are cooked in the boiling water but the tips in the steam. They are also terrific griddled for about 58 minutes. By the way, asparagus has a short season and the imported stuff really isn't very nice. Don't bother buying it until about May next year.
Shallots are onion family like leeks, spring onions and garlic. Chop or slice and soften in oil/butter/oil and butter over a low heat for 45 minutes before adding your chicken. They are a pain to peel, but it's made easier if you put them in a heatproof bowl, pour in boiling water to cover, leave for a few minutes and then drain first. You could cook them whole, but I would recommend baby or pearl onions instead.
Sorry that was a bit of a bunch of instructions, but it is my profession.