As the question says, I would like to know the difference(s) between an Architecture review and a Design review. If someone could explain the two using an example, such as an Architecture review of a network and the Design review of the same network, it would help greatly.
IT Engineering is part of the broader Systems and Software Engineering process, which has well defined stages that are taught in virtually every College and Trade School in the world. In general there are numerous stages in the IT engineering process: requirements, architecture, preliminary design, detailed design, integration and testing, security/penetration testing, staging, production, and operations.
So no, the Architecture stage does NOT include all aspects of design, cost estimation, material/component choice, documentation, and getting everything to work together, but does include planning. All the rest are intentionally separate and further stages of the process. An Architecture many times has an upper-limit cost target defined, plus may have some cost and schedule estimates of most-expensive components or critical-path sub-systems or components. And sometimes may include some preliminary design of especially complex sub-systems of the IT Architecture.
Your IT architecture is a high level design of the overall major elements of the IT system and how they broadly interconnect: telecom, gateways, routing & switching, servers, desktops, remote access, security, etc. Its review will need to answer questions about how the architecture meets requirements (for example, if remote access by employees is needed, how is that accomplished; or if there is a requirement for Internet facing servers yet protected back-end databases servers, what are the elements in the architecture that provide staged separation yet controlled access). It is acceptable to have some unknowns and TBDs during the Architecture review ... ideally as few as possible but still this is a review of the conceptual vision for the IT system, not yet all its nitty-gritty details.
Design Reviews follow the Architecture Review, and there can be several Design Reviews depending on the complexity of the IT architecture. The Design Review will fully describe the exact details of the IT system being proposed or being modified. Some examples include: what specific components (hardware and software) from which vendors and which models/versions (a bill of material aka BOM), what is the BOM's cost details (one-time and recurring), how specifically is the system maintained by the operations staff, what security tools are to-be in-place, how does the security operations staff use those tools daily, what is the routing plan, what is the IP addressing plan, what is the server loading and load-distribution plan (startup, typical, worst-case recurring, surge, and under-attack), what is the DNS implementation (external and internal), etc. Each subsystem of the IT Architecture gets a detail review.
I would suggest you do some reading about IT Engineering and Systems Engineering ... there are all sorts of books available on a decent bookstore about this topic. You can also search online for Enterprise IT Architecture Review checklists and Enterprise IT Detailed Design Review checklists ... some are available online from mostly the U.S. Govt and from Industry Standards bodies like The Open Group.