I really want to put a six-speed free wheel on here and a rear derailleur but I don't know if this bike will accept a RD. Does anybody know if I would be able to add a hanger or if I can do a Direct mount Shimano? It is an older road bike and I am not sure the make/model of it. It is a 26 inch frame with 28 inch wheels.
The derailleur attachment itself is not what's in question with this need. You can use any of the products on the market that are made to do it, like the Fyxation thing or the Problem Solvers thing, or you can adapt a claw hanger RD, or a claw hanger adapter like the Sunrace, or you can have a framebuilder replace the dropouts.
What's in question is everything else about the practicality and cost-effectiveness of converting it, since it will mean doing a lot of other stuff to the bike. Since 120mm-rear-end-friendly contemporary 5- and 6-speed freewheels are almost all nonsense parts, there is a lot of impetus to spread the frame unless you use the Fyxation product, and if you do that then you're locked into their proprietary cassettes. The other ways of doing this all involve spreading the frame to 130mm and realigning the dropouts, then buying a normal road cassette wheel plus cassette to match the new configuration. At that point you can have a mechanically sound track bike with gears on it, but in my experience it's not always obvious that it's all worth it.
I don't have an opinion on the Six Fyx and this isn't a venue for product reviews anyway, but to my understanding it only can ever give you 12-24, which may be a dealbreaker if having much of a gear range was why you wanted to do this in the first place.
No, not easily.
What you have is a "track-end" where the chain's tension is adjusted by moving the wheel back and forth then cinching down bolts securely. There is no good place to attach a derailleur.
However there are some possible solutions.
Fit an internally geared hub instead of a rear derailleur.
More detail at Can I fit a modern internally geared hub on an old bike ?
Make your own custom derailleur hanger, but you'll need to allow for the slot being a different angle, and that the open end is in the other direction.
These are available retail for cheap:
but you'd want something that looks more like this:
The hidden problem is the Over-Locknut Dimension of this frame. If the OLD is 120mm or 126mm then there isn't room horizontally for a wheel and cassette, so you're left with trying to stretch the bike if it is steel. You say its a 26" frame (66 cm) which implies some long chain/seat stays, so this is "possible". If it were a 20" wheel BMX then I'd say its unlikely to work.