I followed him into his room because it says 'seguito'. If the auxiliary verb is a form of 'avere', the participle (seguito) agrees with the object, in this case l' (lo). If you wanted to say 'I followed her into her room' you'd say 'L'ho seguita nella sua stanza'. Not a native speaker but pretty sure this is correct.
This is true. When you elide/contract your object pronoun (lo/la/li/le) you must make sure the object agrees so the listener knows who you're talking about. So yes, "I followed her" would be "La ho seguito" (GENERALLY INCORRECT IN SPOKEN ITALIAN BUT NOT INCORRECT IN WRITTEN ITALIAN) or "L'ho seguita" (ACCEPTED IN SPOKEN, SO THIS IS HOW WE ALWAYS WRITE IT).
However the alternative ending "I followed HIM into HER room", while maybe a bit ambiguous, should be correct and at the moment is being marked as wrong. There would be a number of contexts where it would be appropriate. For example, a little story:
"Io e il poliziotto stavamo investigando l'omicido della ragazza. L'ho seguito nella sua stanza". Or in English: "The police officer and I were investigating the girl's murder. I followed him into her room."
Ambiguous, but it may make sense in context.
In Italian in contrary to English and German the prepositions used with movement or status are the same.
Sono a Roma = I am at Rome.
Corro a Roma. = I run to Rome. OR I run in Rome.
Sono in Sicilia = I am in Sicily.
Corro in Sicilia = I run to Sicily. OR I run in Sicily.
It's the same in this case, unfortunately it's ambiguous. It could be: I followed him into his/her room OR I followed him in his/her room.
Normally it's obvious by context if not Italians would express it in a different way, for example: Mentre eravamo nella stanza l'ho seguito...