Can my phone write NFC tags?
Your phone can read and write NFC tags, but only if the tags are writable. Many are read-only, or are write-once.
Is my phone an NFC chip itself?
Yes, your has an NFC chip, in a sense. It is merely a much more advanced one, as it uses the phones processor to do logic and respond.
All unpowered NFC tags have a tiny chip that responds when powered (the scan itself provides enough electric energy to power a standalone NFC chip). There is nothing particularly special about it, other than the very limited capabilities and very low power requirements. Some NFC chips even do a secure handshake, such as credit card chips; They don't just respond with a preprogrammed response.
Your phone is just as much of an NFC tag as any other. However, you can actually control the response it sends, and you can even disable the NFC sensor to prevent scans.
One big difference is that the phone must be powered on to respond. Unlike NFC tags you are familiar with, a phone does not have a low powered processor capable of responding using the energy solely provided by the scanner. So, in that sense, your phone does not have an NFC tag.
But if a phone has an NFC scanner, it by definition can also take on the role of an NFC receiver (aka tag). It is the same frequency and technology, and all that matters is which end initiates the scan. An NFC tag can only respond; your phone can do both roles.
Why are some NFC tags not supported?
Not all NFC tags run on the same frequency. There are competing standards, and some phones can read certain sets others can't.