Cheapest Travel Possible

It would be a good idea if you put the countries in the order you want to visit them and if you tell us what flights do you want to take (I know they are showed by dashes but I guess you don't want to fly from Spain to France so I think there's something wrong with the dashes) and what countries you want to reach by sea.

In any case, I think you are missing some flights.

You'll have to fly in and out of Burma.
Nepal and Bhutan don't share border and the border between Bhutan and China is closed so you'll have to fly from Nepal to Bhutan or to go through India.

Argentina and Colombia don't share a border either... You could make Argentina-Brazil-Colombia but if you don't want to fly, you would be restricted in Colombia to the area around Leticia...

How do you plan to go from Italy to Britain?

Regulations for foreiners visiting Tibet change frequently so take a look at what are the regulations on your travel dates. Sometimes it's forbidden for foreigners, others you can make it if you get a special permit and go with a tour...

Bhutan requires a minimum spending by day in order to let you in.

I intend to work as I travel.

I assume you know you'll have to get the relevant work visa in order to make it legally. Therefore, I'm going to suggest you to take a look at what are the requirements to get a work visa in some (all?) the countries you plan to visit in order to get an idea of what to expect.

""Please do not try to convince me not to do this or try to tell me it is impossible""

It's hard to convince the impossible..lol
I could pick a thousand holes in this little dream..

I come up with 14 flights that will be needed..

What is your nationality OP ?

Good luck however this trip isn't very plausible.

Getting work in countries such as China and Brazil would mean stealing the jobs from the locals, and hunting and gathering in Papua without a weapon... again you would be taking from the locals, or they would kill you, there are some pretty intense tribes on that island.

Sleeping without shelter could be fine if you are on a remote beach in a hammock, however this emergency cash you speak of would soon be stolen from you.
The idea is good, but you'd need some basic travel experience first, and you'd better choose only 1 country, learn a language, have a skill, let's say teach English etc.

I don't think sleeping on the streets is necessary... Plus, in Britain/Ireland you would definitely freeze! Couchsurfing is free & you can also get advice for cutting costs in that particular town/area/country from the host you stay with.

You'd be hard pressed to do Ireland on the cheap. Finding work would be extremely difficult & "hunter/gathering" means you'd probably end up having to pilfer from someone's land or trespassing which wouldn't go down well.

Ah ambition, optimism and blind niaivity, a great mix!

Have you tried doing this for even a short time in your own country, just to try it out? Even if you sleep on the streets, food is still a major issue.

I estimate your emergency funds will last about as far as Japan. Can't see any way of living without money there. Australia and New Zealand are your best bets for working but you need to be able to demonstrate a certain level of funds to get the working visas in the first place, assuming you are even eligible.

Dream on.

There are Hobos everywhere who live their life as you suggest, many of them do it out of choice, many not. Of course they don't tend to hang out on Travel Forums. As mentioned above you're going to have to catch quite a few more flights than you suggest. Currently it is the Law in Bhutan that tourists have to spend a minimum of $250 a day, though that includes accommodation, guides, and a donation to Health and Education programmes... You can't just enter the country without going through a local operator.

In some places you may be able to find work, even being paid under the table or against room and board but it's going to be a lot harder in other countries. Hunting, even without weapons is illegal without permits (also known as poaching) and could get you into serious trouble and as mentioned above gathering will probably end up being you pilfering from someone's land.

Don't consider the above as me trying to convince you not to do it or that it's impossible, just trying to give you some information about the hardships. The moral issue of sponging off countries like Nepal, Colombia, PNG is a different matter.

He will last 5 seconds in PNG..

I'm wondering how he expects to wash himself..
A begger who stinks aint much fun.

Sooner or later when you smell bad enough and look disheveled enough some countries immigration will make you prove you can support your self and when you cant they wont let you in. Australia for instance if they chose to ask you, you will need to prove you have $5000AU to enter on a working visa. If your really serious about this trip i suggest you go live on the street for a month now in your home town and see how long you last. I doubt it will be very long.