If you sum up independent random variables, then the variance of the sum = sum of the variances.

In general, you can expand your sum expression into all the individual terms (products of a variable with itself and with all the cross-terms) and compute the expected value of the individual terms.

Well, more practically than the two previous answers I would observe that you are looking at scalar products of random vectors, which are product of their teo norms by the cosine of their angle. Take it from there. Let me know if it helps (recommend the answer) or share more details. See attached document which is your document where I have inserted a proposed method

First, you should define a new random variable Zi=XiYi. Then you can try to compute its variance as follows

Var(Zi)=V(XiYi)=E(Xi^2Yi^2)-E^2{XiYi}=E{Xi^2}E{Yi^2}-E^2{Xi}E^2{Yi}

Finally, because Zi's are independent

Var(sum(Zi))=sum(Var(Zi).

As what the rest response said, if X_i and Y_i are independent, it is not difficult. If X_i and Y_i satisfy some other properties, such as subgaussian, you can check the probabilistic bound in Chapter 2 of Vershynin's high dimensional probability book.