Cryptographic Hash Functions

Alice wants to monitor changes to all files on her computer. She uses HMAC and appends HMAC(k, file name || file content) to each file. She updates this value for every change to each file. Can Alice, using the HMAC values in the files, detect whenever the following changes are made by an attacker (Eve)?

1. Content of one of Alices files is changed.

2. One of Alices file is deleted.

3. A new file is added.

Assume that Eve (e.g., in the form of malware) can save and re-use previous valid versions of Alices files and corresponding HMAC values.