Variants of direct product theorems
(1): This question was studied in the paper "Towards proving strong direct product theorems" by Ronen Shaltiel, and it turns out that such a conjecture is false: For example, it could be that $f$ can be computed with probability $0.99 * p$ with size much smaller than $s$, and only the additional $0.01 * p$ probability mass requires size $s$. In such case, when computing $f$ on $k$ instances, the circuit could solve $f$ on most of the instances with size much smaller than $s$, and will need size $s$ only on few of the instances.

(2): A direct product theorem for worst-case complexity is known for formulas and for monotone circuits, but is actually known to be false for general circuits. For an easy example, consider a function $f : \{0,1\}^n \to \{0,1\}^n$ that views its input as a vector and multiplies it by some fixed $n \times n$ boolean matrix. Then, computing the function $f$  may require size $n^2$, but computing it on $n$ instances can be done much faster than $n^3$ using a matrix multiplication algorithm.
You can find a thorough discussion of this subject in the book "The complexity of Boolean Functions" by Ingo Wegener - see Chapter 10.2 here: http://eccc.hpi-web.de/static/books/The_Complexity_of_Boolean_Functions/.