Spatially-coupled (SC) codes are a class of low-density parity-check (LDPC) codes that have excellent performance thanks to the degrees of freedom they offer. An SC code is designed by partitioning a base matrix into components, the number of which implies the code memory, then coupling and lifting them. In the same system, various error-correction coding schemes are typically needed. For example, in wireless communication standards, several channel conditions and data rates should be supported. In storage and computing systems, stronger codes should be adopted as the device ages. Adaptive code design enables switching from one code to another when needed, ensuring reliability while reducing hardware cost. In this paper, we introduce a class of reconfigurable SC codes named rate-memory-compatible SC (RMC-SC) codes, which we design probabilistically. In particular, rate compatibility in RMC-SC codes is achieved via increasing the SC code memory, which also makes the codes memory-compatible and improves performance. We express the expected number of short cycles in the SC code protograph as a function of the fixed probability distribution characterizing the already-designed SC code as well as the unknown distribution characterizing the additional components. We use the gradient-descent algorithm to find a locally-optimal distribution, in terms of cycle count, for the new components. The method can be recursively used to design any number of SC codes needed, and we show how to extend it to other cases. Next, we perform the finite-length optimization using a Markov chain Monte Carlo (MC$^2$) approach that we update to design the proposed RMC-SC codes. Experimental results demonstrate significant reductions in cycle counts and remarkable performance gains achieved by RMC-SC codes compared with a literature-based straightforward scheme.