Are there any Intel/AMD desktop processors that support weakly ordered memory or this is a feature of a server setup with multiple processors?
I'm not sure if this is what you're after, but IIRC there have been some architectures that had weakly-ordered memory accesses in that they could be ordered arbitrarily, and you had to insert memory barriers to ensure a particular ordering.
Modern processors use what's called a "load-store queue" that hides memory reordering, making it look almost as though it's happening in program order. Reads are often reordered (but with some care), writes can be made out of order but are committed in-order (although multiple writes to the same location are consolidated), and reads and writes are reordered wrt each other only carefully and speculatively. The latter is called "hoisting", where a read is performed speculatively ahead of a write (that appears earlier in the instruction sequence) and may be canceled (like a mispredicted branch) if it turns out that preceding write would have affected it.
Also, if memory is marked as uncached, then CPUs generally infer that to mean it's I/O space and perform no access reordering. x86 and SPARC are like this. However, PowerPC will still reorder reads to I/O memory space, and we have to use the EIEIO (ensure in order execution of I/O) instruction to force a particular ordering. IIRC, we also had to use memory barriers on PA-RISC and Alpha. Moreover, there are memory barriers on x86, but I'm not familiar with their use (possibly to ensure ordering of accesses to cached memory space).
You mention multi-core systems. In general, elaborate cache coherency protocols are employed to make all memory access appear to conform to certain interleaving rules, such that accesses hit the last-level caches and main memory in an order that would be possible if there were no caching.