You didn't mention what platform. Is it Blackberry, Android, Window Mobile or J2ME?
If we get a programmer to create a mobile phone game (java) - can we sell it easily from a website direct to consumers.
Depends on your criteria for easily, but it is doable. You want shopping cart software that will either protect the download until after purchase, or assign a unique key at purchase to unlock the full version. I did my own custom checkout, but there are probably some shopping carts that work.
Can people download something on a PC and then put it on their phone easily?
Yes, in most cases. Windows Mobile can. Android can with a few more steps. IPhone can't in general - the market is locked down. I haven't done J2ME or Blackberry but I believe so, since Handango and other stores carry them.
If you create the proper Java container then yes, you can have your application downloaded directly to the handset via the mobile user being directed to a URL that initiates the download. This is standard course for Symbian application distribution, and various Blackberry (Java) apps are also distributed this way. Check out fring to see how they handle instand download of their app to a variety of handsets to get some ideas.