Assuming you cannot have an empty domain (otherwise, all sentences would be vacuously true, and thus true in a finite domain after all), you need at least one object. This object needs to relate to something, but that something cannot be itself, so you need a second object b. b needs to relate to something as well, but that cannot be b itself, and also not a, otherwise by transitivity b relates to b after all. So, b needs to relate to an object c. Etc.
And to make that 'Etc.' a little more hard: Note that any model can never have a cycle of aRb, bRc, cRd, ... yRz, zRa, for once you have such a cycle, then by transitivity, all opjects in that cycle end up relating to each other, including themselves, thus contradicting irreflexivity. But if there is no cycle in your structure, then it is possible to 'line' up all objects from 'left' to 'right', such that any R relation goes from 'left' to 'right' as well. But with a finite number of objects, that means that you end up with a 'rightmost' element ... which therefore does not relate to anything, and that contradicts the ∀x∃yRxy
So, you can indeed not have any finite model
For c) No, you can't just negate the statement to force a finite model. We showed that in order to satisfy the sentences, we need an infinite domain, but all that that means is that in a finite domain, the negation of the statement is indeed true. However, that does not rule out having infinite domains where the negation is also true. Indeed, it is easy to create an infinite model of the negation: just have infinitely many objects, and have them all relate to each other (i.e. define R to be reflexive)