Hello From Australia

Aus license valid for 90 days. You need a certificate of residency - says where you live - to put the bike in your name. If registered in Thailand, and in your name, make sure that the insurance you buy will cover you with an Aus license. If not, you will need a Thai one. Can get one easier if you have a Aus license valid for motorcycles. To take the bike out of the country you need it to be in your name, or have the proper paperwork from whoever owns it - bank, friend, etc. To travel to Laos you need a vehicle passport - purple book. Cambo and Malaysia just need the green book translated into English. Would you trust your friend if you buy the bike, put it into his name, and do that ? Will save you effort, but all doable either way

Purple book is no longer available for motorcycles at most motor transport offices and not needed for Laos, from recent experience. No translation required for Cambodia. The residency certificate ( address letter ) is available from the immigration office of the province where you have the address you want to use.You need back up documentation to get this letter, this might be a lease for a house, blue house book copies, ID copies, your passport copies, entry card copies, passport photographs . Sometimes tourist police issue a similar letter but some motor transport offices don't accept this (ie, Korat ). If you can get the letter, get 2 copies issued at the same time, one for the licence and one for the registration. The letter is valid for 6 months.It is "free" ie. no fee, but most likely you will never get it without a donation of some baht ( can be 500 or more) . Legally (not always in practice) you need a long term visa (90 day or more) for the registration and licence . Be prepared for some interesting logic and unexpected delays when registering a motorcycle or any other government related paperwork in Thailand. The trusted friend route is probably the easiest for time restricted travel. Make sure the letter giving you permission to take the motorcycle out of Thailand by the "legal" owner has all their and your details on it and has some tax revenue stamps attached (makes it a legal document). This is all from personal observation and may vary greatly from location to location in Thailand. Good luck and be persistent.

Harri has summed it up pretty well..

My 5 cents worth below.. From what I know or heard from friends.
Open to correction or additional input as things change all the time.

"Australians get a 30 day visa on entry - is it possible to get a bike with this visa? - as I am unable to obtain the longer, 3 month visa. "
You need to be on a Non Imm B or O with appropriate supporting documentation.
Non Imm B a work visa.. which requires you to have work permit etc.

Non Imm O.. can be retirement, student etc etc.. many other categories.
All of which you will need to jump through some hoops and show money in the bank etc..
From your original post.. These don't seem plausible for you..

"Does anyone know any bike shops that specialize in selling to foreigners and can stream line the registration process? The buying a bike page mentions that there a bike shops that can get it done without the long visa. Does anyone know of a specific bike shop that can confirm that they provide this service? I would be willing to pay a premium for any help a bike shop can provide. "

Unless the shop is greasing the palms of the officials.. It wont matter which shop you go to.
Some may try to assist to make the sale.. others may not want to be bothered.
Different provinces are more lax than others.. eg.. Bangkok you have no chance on your tourist visa.
I have heard from a friend it is possible in Chiang Mai.. But you will need a letter from the hotel / landlord or something similar from where you are staying ( Can't remember the name - May be Moto Rex can help out here on what he needed ).. No one really likes to do this.. In case you rack up bills or do something else that will bring the authorities around to hassle them..
So unless you have a good relationship with someone already.. Probably unlikely you will get this letter.

"I have a friend living in Thailand - what kind of forms do I need to obtain to use his address to register the bike? I read somewhere I needed a statutory declaration- is that all? "
Covered by Harri.. Also on this forum some where..
The friend buys and in his / her name , then there is a temporary ownership transfer that needs to be filled out to cross the border.
More info in the thread on the forum. But some times they can be a bit funny and give you hassles when leaving the country.. usually OK.. but have heard of cases..

"Are there any Auzzies out there who have done this recently and can offer advice? "
Things change here monthly, weekly, daily.. also depending on the last official you spoke with.
So while people here can advise.. you wont really know until you go through the process yourself.
- But pretty confident you will not get a bike in your name on 30 day visa waiver unless you have that letter and accommodating person at the LTA in a province like Chiang Mai..

Hope that helps.