Massless particles

Sort of. It's true that, in our current universe, the only particles that are massless (photons, gluons, and gravitons--the last one has not been detected but is believed to exist) are force carriers. However, in the early universe, that was not the case: all of the Standard Model particles were massless. As the universe cooled, a phase transition took place that caused most of the Standard Model particles to acquire mass through their interaction with the Higgs field. So there is no fundamental requirement that massless particles must be force carriers; it just happens to be that way in our universe because of how it evolved.

No. First, there are force carriers in our current universe that are not massless--the weak bosons (W+, W-, and Z). Second, as above, what we think of as "tangible particles" were massless in the early universe.