While I understand that LinkedIn is currently on the decline amongst the research community, I think it's still fairly well used. I graduated from my PhD 2 years ago, and many of people graduating in the same year use LinkedIn as one of their professional profiles, as do some of the "seniors" I know. Things are shifting to websites (like ResearchGate or similar) specifically targeting academics which mostly show to list and organize publications.
That said, I keep everything in my LinkedIn profile. I never took anything out - stuff like presenting my faculty at the University day in my BSc. is still there. And I don't provide a link to it with my applications - my name, plus basic info from my CV (current/last position) is always enough to find and correctly identify me on LinkedIn if anybody wants to look it up.
In the CV, the things I would typically include:
- past education and experience
- technologies / tools used or learned per project / study programme / job
- publications (+ best paper awards)
- student exchanges/internships
If you are applying to a graduate school, and in general in early career, I'd include the list of all of the above, exhaustively. As a rule of thumb, it should not be longer than 2 letter pages (so if even printed, fits on one sheet of paper) and have just pure "facts" (a short statement of purpose of 2-4 sentences is lately encouraged).
As you grow more senior you remove things (my BSc and MSc internships will go soon, PhD internships a bit later, any pre-PhD honors and awards are gone, a selected publication list instead of a full one happens as you publish more), while still keeping it at 2-3 pages long at most.
The one document I would expect to find with an application that you don't seem to mention is a cover letter (referred to in some languages as literally "motivation letter"). This is the place to:
- elaborate on the points of your CV especially relevant to your application
- draw conclusions and connections between your experience, research and the position you are applying for
- state your motivations, why you think you will be a good fit and why you think the institution / position will be a good fit for you
- mention any cool things that can not go on a formal CV, e.g. ongoing collaborations
So to summarize: think of a CV and cover letter as a unit. One does facts in a concise list or bullet point form, and the other one relates those facts to your experiences and makes them relevant to the application. LinkedIn is just an additional, professional profile, which people can find you based on your name and where they can look up more detailed info if they like, like links to and summaries of past projects and publications which are too clunky for a CV.