To identify a card, you power it using a carrier wave on its frequency and wait for answers. If there is no answer, you switch to a different frequency and protocol and so on until you get communication from the card (you may need to use multiple readers as different frequencies require different readers). Using your phone is accurate, I mean, if the phone can talk to the card then at the very least the card is using a frequency and protocol that's supported by the phone's reader.
Now, let's assume you positively identified a Mifare Classic card and you need a reader to attack it. Any LibNFC-supported reader should work, but I suggest you look for NXP PN53x-based readers, they are pretty cheap (chinese, and I'm not sure the NXP chip is genuine but they do work) and work perfectly with LibNFC and all applications that use LibNFC to talk to readers.
For software, you first use
mfcuk to get at least one key, then you pass that key to
mfoc which uses a different (and faster) attack that requires the knowledge of at least one key (you can also try default keys if the people who manage the card system you're attacking were really idiots). These software use LibNFC to talk to the card and are thus compatible with any reader supported by LibNFC.
Yes, once you have the key, any device can read and write to the sectors protected by that key. Note that there can 2 different keys per sector, and each sector has some ACL bytes that describe the permissions for each key.
Note that Android devices with NXP chips can directly crack Mifare Classic cards using NFC-War.