Firstly, welcome to Canreef
The coral in the first picture looks like Green Star Polyps (Briareum sp). They grow very quickly and for this reason, some people don't like them. I'm not sure what the stringy things are--could be various things (digitate hydroid [sort of not too bad] or Vermetid Snails [generally considered reef safe, but can multiply and become pesty).
The coral in the second picture looks like Kenya Tree (Capnella sp.) and is reef safe. Again, though, it can grow very quickly and drop little "buds" around the tank, so you'll suddenly find them sprouting up everywhere! That's fine if you like them.
I can't make out what the "mushroom" type coral is.