TandemGeek

Your older Burley would be a bit more prone to frame deflection than many more contemporary, major brand-name tandem frames as larger diameter, stiffer tubesets are used for the boom tube and elsewhere in the frames. However, unless you find your tandem is ghost-shifting under heavy pedal loads or throwing the timing chain, the frame deflection is nothing I'd lose sleep over. It could also be an issue if you're trying to shave a few ticks off the clock on your time trial times as it's conceivable that you could be losing some efficiency, but that too seems to be a non-issue for you (and most every other tandem team). Of course, if you've got any significant frame wind-up chances are the rest of your frame isn't all that efficient either: again, if you're not racing and the tandem otherwise meets your needs and expectations who cares.

As a rough gauge of how much deflection may be occuring, ask a riding mate to look at your timing chain when you're pedaling under a heavy load and let you know how much it's sagging compared to when your tandem is at rest. A certain amount of the sag comes from the chain going into tension but excessive difference between how much a timing chain sags when static versus under load does suggest there's some frame deflection going on.

It appears as though you're still running the original 7 speed Shimano HF05 (STX/LX) tandem hub which, admittedly, isn't top-shelf. They were more than adequate with 12x28t cassettes, but the 34t models were the kiss of death for larger teams using the HF05, as well as the 8 speed Shimano HF07 (STX/LX). Shimano solved it's tandem hub durability issues when they introduced the [Delete] HF08 (XT) grade 8/9 speed hubs which remain one of the best-value tandem hubs you can get. They're not sexy, but they are durable... so much so that Tandems East and others use them as the standard hub on their tripets and quads. In fact, the internals of the HF08 are what Shimano uses for the hub of their Shimano/Santana Sweet 16 tandem wheelsets and I believe they are still what you'll find inside Shimano's XT hubs. So, my point in running through all of this history is to suggest that if you have concerns about your rear hub and the rim is otherwise still in good shape you merely have the wheel rebuilt, perhaps with new spokes, using the HF08 and also upgrade your shifter and derailleur to 8 or 9 speed at the same time... the latter readily available second hand or for not a lot of $$.

Not sure I fully understand what has been split and welded -- the axle? -- but given that you're using a right-side drive any standard MTB square taper (JIS) crankset will work; choose your finish to match or contrast with your rear cranks which can also use standard MTB models. I'll leave the re-engineering of your front BB to you and your helper. While 97mm is ideal, you could go wider if needed so long as you don't create an interference problem between the timing chain and the chain on your middle ring.