Multithreading: do I need protect my variable in read-only method?

Yes, unless you can guarantee that changes to the underlying variable counter are atomic, you need the mutex.

Classic example, say counter is a two-byte value that's incremented in (non-atomic) stages:

(a) add 1 to lower byte
    if lower byte is 0:
(b)     add 1 to upper byte

and the initial value is 255.

If another thread comes in anywhere between the lower byte change a and the upper byte change b, it will read 0 rather than the correct 255 (pre-increment) or 256 (post-increment).

In terms of what data types are atomic, the latest C++ standard defines them in the <atomic> header.

If you don't have C++11 capabilities, then it's down to the implementation what types are atomic.

Yes, you would need to lock the read as well in this case.

There are several alternatives -- a lock is quite heavy here. Atomic operations are the most obvious (lock-free). There are also other approaches to locking in this design -- the read write lock is one example.