I don't know of any free sources for a dynamic list of symbols. Note that symbols change all of the time, so just having a list of symbols at a particular point-in-time will be difficult to match with a price database of symbols that have "current" symbols.
REITs are a subset of normal operating/holding companies. They differ from these by electing to be taxed differently to a normal investment company. There are various qualification rules to be considered a REIT:
Not all REITs have always maintained this designation across their lifetime. Have a look at the REITWatch newsletters - probably the most comprehensive source for such information but you should cross-check the REIT designations with more sources as sometimes it took a while for that newsletter to add/remove valid/invalid REITs.
For example, Crown Castle International (NYSE:CCI) gained REIT status in late 2013 but the prior 1998 through to late 2013 it was just a normal company.
A further anomaly is that a company may think it is operating as a REIT but then has to retrospectively alter its designation once the SEC declare it is no longer a REIT.