At the Altar of Rites, does one have to sacrifice material - Arcane Dust, Forgotten Souls, etc. - or will actual items (swords helmets, etc.) work? If one sacrifices material, does it have to be higher-level, scarcer stuff; and, if not, does rarer stuff provide more benefit?
Finally, just how does one know which circles to activate when and in which order?
I hope you can forgive my no doubt stupid questions, and will accept my thanks for any help you might give me.
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Also a guide here:
https://maxroll.gg/d3/resources/altar-of-rites-explained
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You need whatever it asks for. If it asks for mats, you use mats. If it asks for gear, you use gear. One thing to remember is, if it asks for gear, it doesn’t work automatically (like ancient puzzle ring) as you may have more than one. When you do those, a hammer will appear and you must pick which one in your inventory to sacrifice.
Nope. Just what it asks for. If it says a specific helm, it doesn’t need to be ancient, regular works just fine.
No extra benefit.
There are a lot of guides, but basically it breaks down to what your goals are. I took the branch that got me to super potion power first (bottom right). You want to unlock the whole thing anyways to get double primal, so go with what you think you need. MOAR power is always good. The utility boosts are very, very good. Save defensive stuff for last, since who the hell cares about defense.
Yeah, the only thing I suggest on this is unlock the double bounties node quickly. After that, … who cares. Make sure when you’ve completed it all that you actually click the final node to get double primals 
It is rumored that Altar has magnets inside of it. A total gizmo.
There are guides out there for Altar to optimize its benefits along the path. Eventually you will unlock everything. To shortly summarize, with each sacrifice the cost of unlocking the next node increases in grade value, so it’s important to unlock the most crucial nodes and unlock the least important ones later if you want to optimize your journey. It doesn’t have any effect on the cost of the next node unlock which order you unlock them, it always increase in value gradually and rarity for another challenge.
Nodes only become available for unlocking when you unlock the closest spreading node pointing to it. For example; potion powers only become unlockable with Primal Ashes and they become available when you unlock the nodes closest to them, just like any other node.
As an exception; middle belt of three nodes, don’t require anything above them to be unlocked once you unlock any of the adjacent nodes from that belt. Last node at the very bottom, will only become available to unlock once you achieved to unlock the whole board, and won’t require any additional cost.
Late edit: a correction.