How to compute secure sum using secure multparty computation? Suppose there are three voters $P$, $Q$ and $R$, and each will vote only on one candidate out of $X$, $Y$ or $Z$, with a 6 bit vote vector corresponding to $X
$, $Y$ and $Z$ respectively (with 2 bits each for a candidate. E.g., $( 01, 00, 00 )$ signifies a vote for $X$). 

How can any voter find out the total number of votes cast for each candidate without disclosing their preferences by the use of secure multiparty computation?

Each 6 bit vote vector is encrypted and sent to the some third party who will then take the product of ciphertext votes and then decrypt the product to yield a bit vector which shows the total number of preferences received by each candidate (Paillier cryptosystem).

What homomorphic property must be held by the cryptosystem in the plaintext and in the ciphertext space to satisfy the objective?

Finally, I need some theoretical explanation finding secure sum using SMP (with and without random seed) and SMC to sort the numbers.
In general, there are many techniques for secure multiparty computation. However, the problem of voting is much broader than this, and if you are really interested I suggest looking into the voting literature in depth.

One of the big questions is the model: typically, you cannot run a regular MPC protocol since that requires interaction among all the parties. One solution (which is an old work, but one that I really like), separates between an election service provider and the actual election being held. Actually the work talks about auctions, but a similar idea can be used. See Privacy Preserving Auctions and Mechanism Design.

For more modern work on elections, see Helios (which is the scheme used by the IACR). I suggest looking at Wikipedia - end-to-end verifiable voting for some sources, but note that a lot of research is needed. In general, secure tallying is a tiny part of what is needed for a real election (and the easiest to solve).