They can only see / track stuff that is publicly available, plus what you give them in your paperwork.
Once you are a registered student or employee, they can search your possessions / locker / room / computer and monitor your internet use while at college / work..
It's in the fine print of virtually all contracts, that you give permission. So, they do not need warrants.
You ask to be at that place, it's a condition of them accepting you.
And note that fake information or breaking the rules on facebook or any other web site is technically a criminal offence - it's an "unauthorised access" offence under computer misuse laws.
They may not bother to prosecute if you are not doing anything malicious but they are likely to ban you from the site (under any ID) if they notice - possibly forever.
And as a point of note, trolls do get prosecuted; something like 2,000 of them in the UK alone last year.
Be careful and follow the rules, it's not worth possibly ruining your entire life by getting a criminal record.