Is the sum of all the wealth in the world a constant value?

No! global world report estimates a 40% increase in wealth worldwide by 2018.

To make the answer obvious, consider two times far apart.  We have more people with more stuff (total wealth = number of people times stuff per person) now than we did in in, say, 500 BC.  Therefore, it is clear that total wealth is not a constant.

All you have to do is think of how poor the relatively few people there were 5  centuries ago and compare it with how many more relatively wealthy people there are now. It's 1 billion people living only until the age of 20 or 30 years old in grinding poverty in 1500,  versus 6 billion, living to age 70, many of whom are fabulously wealthy today, especially compared to 5 centuries ago.

No, but a lot of this depends on your definition of "wealth".  But we might just say that its a human concept and that on average humans value certain things, or states of matter, more than others.  For example, most humans would say that carbon in the form of a diamond is more valuable than a lump of coal. Although economics is fixated on ideas such as labor, capital and land, more fundamental is simply the idea that energy can be used to transform matter from less valuable forms to more valuable forms of matter.  And energy itself has value, but again this is somewhat determined by what form it comes in and whether it can be used in that form or needs to be transformed itself. Wealth is created when energy is used to transform matter from less valuable forms to more valuable form minus the value of the energy used, or when energy (like the sun) is converted into a more useful form (like electricity).  Or for example, when a crop is grown, turning seeds into food. Loss of wealth and entropy are related concepts.  Loss of wealth can happen when valuable things decay and when energy is lost to unwanted heat.  Or things are wantonly destroyed through war or other destructive acts. So as long as we have new energy -- from the sun or elsewhere -- it is possible to create additional wealth.  But wealth is also constantly being lost due to decay, destruction and waste.   There is no requirement of equilibrium here, because its not a closed system.  Wealth can increase or decrease either as a whole or in particular parts of the world.

In purely monetary/economic terms, no. As described by everyone else here, market "values" certain things higher than others - for e.g. for a viable manufacturing process value of finished goods > raw materials that go into it + the process involved Hence every time something is manufactured, we are creating new wealth, at least as seen from the our point of view (at the very least for those who are benefiting from these conversions). However, if you measure wealth in terms of more basic terms (for e.g. is clean air/water = wealth? Or Energy = wealth? OR Are mineral deposits wealth?) the answer could be a bit less clearer. It will also depend on what we value more - for e.g. certain if someone values untouched forests and being closer to nature more than health-care and prolonged life, family values more than technological advancements, then they might perceive overall wealth as actually decreasing in today's world, even if some others perceive wealth as increasing.

One example: Because value is subjective, a voluntary exchange is made when the final result will be that each and all of the parties involved end up with something of more value TO THEMSELVES than what they started with. Total value has increased without an increase in physical stuff.

answers this question very well: http://www.paulgraham.com/wealth.html The full essay is very interesting, but if you want to go straight to the point "The Pie Fallacy" section is what you are looking for.

A few people have said wealth is a subjective quantity. If we define it as matter structured in a complex way to provide usefulness to humans, then it's clear enough that wealth can increase. Increasing wealth means building complex structures. In the process we consume raw materials, including energy. Wealth trap number one: it's alarming how quickly wealth can turn into no-wealth. A written-off car is still complex, but of no use. Manufactured plastics in landfill are complex, but no use. The raw materials and energy used to create them are still gone, and so is the wealth. Another way to define wealth would be to include potential wealth. Raw materials and energy constitute potential wealth, which can be converted into actual weatlh. Even busted cars and ripped plastic bags could conceivably be converted into wealth again, given enough energy. Putting it like this raises the cost of producing the wealth. Cost like wealth is subjective. If the perceived cost is higher than the the resulting perceived wealth, it doesn't rate as wealth,  according to the pertaining perceptions of cost and wealth. Wealth trap number two: not only can wealth suddenly become non-wealth, but variations in economic conditions can alter the cost/wealth balance. Existing wealth can become less valuable than the materials used to create it. When there are no more fossil fuel reserves, this effect may become quite noticeable. If wealth includes potential wealth, we can reduce our wealth by not carefully conserving our raw materials and energy. The question seems to refer to the laws of thermodynamics and the conservation of energy.  The Earth is not a closed system, though. Energy keeps arriving from the sun. If we could work out our budget for energy, and ways to recycle all the complex landfill back into useful raw materials within our energy budget, we could continually maintain or increase our wealth. The budget only includes renewable energy. The one-off fossil fuel energy bonanza, due to end this century, isn't part of it. We don't get to hang on to our wealth if we can't preserve the economic and social conditions that allow it to be perceived as wealth. To take it to absurd lengths, in the event of a zombie apocalypse, all our wealth disappears, unless the zombies are really into dead iPads. In a more believable scenario, we experience a tremendous collapse in wealth every time the power goes out. When we can't run our power stations, wealth isn't going to be increasing. And there is no wealth whatever in financial instruments, unless they refer directly to something of use. A dollar is a promise, and it's no use if it isn't kept. The derivative disasters of the last few decades underline this point.

Yes, in the long run, a constant value of zero. Before the advent of life everything was worth nothing, and after life is dead and buried, everything will again be worth nothing. Economics is the art of accounting for the various bubbles between these two end points. It all depends upon what you count and how you do the counting. In particular, the current monetary regime (compound interest) capitalizes past gains and discounts future liabilities - thus creating positive present values.