Small nucleus emission from a larger nucleus Like alpha decay, is there the possibility of a small (n,z) nucleus coming out of a large (N,Z) nucleus? Why lithium and beryllium don't decay out of big nuclei as helium does ?
 A: The decays you are considering are determined completely by the energetics. The alpha decay occurs so readily because the alpha nucleus is so tightly bound. Lithium and Beryllium are much less tightly bound so the energetics are less favorable. You can explore the possibility of such decays by looking at the masses of the nuclei involved. If the mass of the initial nucleus exceeds the sum of the masses of the product nuclei, the decay is possible.  The extra energy goes into the kinetic energy of the products.
A: Yes, this is possible. It is the case of $^{223}\textrm{Ra}$ for instance, which can decay through an $\alpha$ process with a lifetime of $\sim$ 11 days, but also through the emission of a $^{14}\textrm{C}$ nucleus. However, this decay mode is extremely disfavored (branching ratio $\sim 10^{-9}$).
There are two factors at play here. One is energetic, because the height of the energy barrier sets the amplitude of the tunneling process.
Another factor is that it is much less likely for a nucleus larger than Helium to form in the large nucleus and escape. This increases the lifetime for this mode by several orders of magnitude.
References
[1] Lund/LBNL Nuclear Data Search, http://nucleardata.nuclear.lu.se/toi/nuclide.asp?iZA=880223
[2] Introductory nuclear physics, Kenneth Krane, section 8.4, "theory of $\alpha$ emission".
