In process of replacing head gasket. Is there a certain way the studs go into the block? One end of the stud is flat and the other is rounded. What is the purpose of the lead washers?

Flat end goes in the block. Those threads are rolled to larger pitch diameter to fit tighter than the nut end. You may need sealer if any are through hole into the water jacket.

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I would suspect that the "flat" end of the stud threads into the block. Is the thread length on that end of the stud a bit shorter? Also, the threads on the block end are designed for a closer fit than the standard thread on the nut end. Likely, that "fit" wears some over time with every removal and installation, but in theory when being installed correctly it will take more effort to screw them into the block vs. the other end (incorrect) end.

As to the lead washers, their purpose has been argued about on here for years.

The consensus seems to be that they provide a sort of "lube" to prevent the steel nuts from gouging into the cast iron head, making for a more smooth/accurate tightening/torquing of the nuts.

The flat short end goes into the block. The threads are not cut as deep or tapered ? so as to help them tighten up in the hole before bottoming out. Lead washers will aid in sealing and lubricating the nuts despite all the people on here that will say otherwise. Be sure to use them just as the factory did. FWIW: Now on models like the 520,530,620,630 from what I have seen in the past If I had the head off of one that I had no history on I'd pull the pistons out too. I have seen several with broken top piston rings. One was so bad it worn a hole up to the top of the piston and a piece of ring was hammering against the cyl. head. At the very least I'd surly mic the cyl. bore to see how much wear it has.