Are the Feynman diagrams still applicable to more than two particles interactions? This would be very helpful in modelling many-body scattering cross-sections.
Feynman diagrams can handle any type of interaction you like, you simply have a different kind of vertex. However, very high point interactions are often irrelevant in the renormalization group sense and so contribute little to low energy scattering processes.
What sort of many-body scattering did you have in mind?
Are there m ingoing and n outgoing particles of a general "interaction blob"? Or are there really m+n particles meeting at one vertex?
The former one is simply a higher-oder process which is typically highly supressed by renormalization group and phase space. The latter one is rarely realized in physical models. You can get something like that in f(R) gravity theories (which some people find rather strange as a QFT) or in non-linear sigma models or effective modelsfor the string interaction, which are to be considered as approximations only