Is it secure to use TOTP with same key and different time-counters? If I use TOTP to generate codes with different "expiration" time, using the same key for HMAC, will I create some kind of weakness?
Yes, that's fine where TOTPs are generated in accordance with RFC 6238; this is because the time is combined with the secret before generating the HMAC. This means that it isn't practically possible to derive the time (or information about it, such as the time difference) based on the key alone because of the Avalanche effect in the hashing algorithm that is used.