Can an iPhone connect to a projector to show pictures?

That would depend on how new the Projecetor is.

It can show pictures in the "slideshow" mode. The iphone uses video out for very few things. It also must be plugged in with an adapter through the bottom of the phone. Video's all work, YouTube works, and slideshows. To date there is only one app that uses video out called picozoo. This manufacturers website has a "support/faq's" page that might help you.