Landing internships

How do you find companies/mail ids to whom we can cold mail?

Voilanorbert is good tool. Rocketreach too

Given that the number of characters are limited when sending a message, how to formulate a good intro within 300 characters? Or do you mean sending a message after connection request has been accepted? Could you share an example?

Ideally it should be when they accept your connection request. But you can also add a short crisp bio( your skillset, and why you interested in the profile) while sending a request. I'm not sure if interships have job IDs but if a company has those better include the IDs in your message to people as no one will be willing to first search those for you and then go to the trouble for referring you. Also look for recently funded startups (accross the world) and reach out to there founders and try to impress them in giving you a intership. It does work but is hard. You can also reach out to the recruiter mentioned in any intership jobs you see in the jobs tab in LinkedIn (this slightly increases your chances of atleast having your cv being read. Also shoot out multiple connection requests for a particular company someone is bound to accept. Just be patient and always be humble.