it's as broad as it's long really. You could either buy a bus ticket to Cambodia or Laos. That's the cheapest onward transport. Or you book a flight to Vientiane (pretty cheap with Air Asia) and get a visa there.
you don't need a visa to stay in someone's house and occasionally walk their dog - other than a tourist visa if (on a UK passport) you plan to stay longer than 30 days...
Yeah, it requires letter writing, but it can be done (I'm a dual British national (though British is very much my second nationality) and I enjoy frustrating (and complaining about) their bureaucracy because I'm sick and twisted like that. So again, possible, though perhaps not even necessary in this case. Apologies for digressing...
Tour guide - illegal. Someone from your company based in the UK who comes to Thailand for a short time to do some 'shepherding' under the overall leadership of a Thai tour guide? Grey area
I have done it ;). If you write them a stiffly worded letter saying you need to travel, and insist firmly enough with the 'cannot' people at VFS, it can be done, despite what it says in their rules....
You don't need a visa for Thailand based on that itinerary. You can buy a visa on arrival in Cambodia. I can't remember what the situation is for Aussies in VN these days but just go on Wikitravel and look
yeah it's different when you go buy car (they still have to pay, but it's presumably bundled into your fare). By bus you have to get off the bus and pay 11k kip