Do you believe welfare is a right?

Absolutely.
Think about what kind of rights we need in the world. There are two kinds:
Rights for people NOT to do things - my right not to be murdered.
Rights for people TO DO things - my right to food and shelter (and other forms of welfare)
Now if you believe in any rights at all, there is no good reason to only believe in the first kind of rights. All rights are bossy: they exist to impose moral or legal rules and to make the world a better place. In a sense, all rights are welfare rights.
The only question is what kind of welfare rights we have: what precisely are we entitled to - which is equivalent to also defining what we owe each other - what our moral or legal responsibilities are to each other.
Of course you don’t need to believe in morality or social responsibilities - but in a sense - if that’s your real view I - and any other rational or reasonable person - don’t need to take the slightest heed of you or your opinions. You’ve abandoned the human family.

Will welfare eventually run out?

No - welfare is just that part of our own time and energy - the time we spend supporting each other.

Is charity more effective than taxation for improving society?

No, because:

• People never give enough freely
• People give it to the wrong things
• It deludes those who give that they are doing more than their duty
• It encourages those who receive to be grateful for handouts
• It is entirely undemocratic and opaque
• It is inconsistent with human rights and international treaty obligations
• It enables elite power and corruption
• That will do for now, but I could go on…

Can welfare be beneficial to the economy?

Yes.
Welfare is the economy. Welfare - faring well - is what an economy exists for.
It is also good for the economy to provide income security - which tends to direct money towards people who make good decisions which benefit themselves and their community.
Of course not all such public spending is good - corporate tax breaks, tax credits and allowances - welfare for the rich - is rather bad for the economy: it locks money into savings and the kinds of lazy investments which undermine or corrupt the health of the economy.
If in doubt, give money to the poor - that money tends to always get recycled to the benefit of everyone.

What created the"welfare state?"

The inevitable dislocation, poverty, inequality and expropriation caused by the agrarian αnd industrial revolutions. 
In other words - when we strip people from the land, leave them dependent on the fluctuations and crises of modern capitalism, we need to create mutual social security - the welfare state. i.e. we need to take care of each other.

How can people reasonably advocate for Universal Basic Income? Wouldn't this just cause inflation and damage to the economy?

As an advocate of Universal Basic Income (UBI), I do think it is important to consider the broader economic impact that such an idea is likely to have. One fear is that UBI would cause inflation.
I think that this fear comes about by imagining UBI being implemented as follows:

1. Everyone has exactly what they had before
2. In addition, everyone gets a UBI
3. All of this extra income is converted into extra spending
4. There is no new production, and so all the extra money is just spent on increased prices for what we had already

No serious advocate of UBI thinks this is how UBI should be implemented, and there are also reasonable grounds to assume that many of these assumptions are highly unrealistic. Nevertheless, there are things we don’t know until we try. The reasons that I think such an inflationary effect would be unlikely include:

1. UBI is really a redistribution of income - not additional income. For most people, it would be new income, because most people would benefit. But the wealthy would lose income with a UBI, probably through taxes or more indirectly.
2. It is likely that people on lower incomes will spend more and reduce their debt. This is basically good for the real economy, whereas less money would be available for speculative investments, monopolisation and uninvested savings. So UBI is likely to make the real economy more efficient in meeting needs and encouraging productive activity.
3. Many economists argue that the current economic system is leading to ‘demand failure’ and that there would be much more real economic activity if there was a new surge in demand. That is, new spending - if it was driven by people’s choices and preferences - would create more wealth (not increased prices).
4. UBI is variable. There may well be a level of UBI which leads to inflationary consequences, and sensible societies can therefore reduce the level of UBI if they start to approach an inflationary (or meaningless) level of UBI.

Certainly none of the actual experiments with UBI have led to inflation: instead it has simply led to improvements in the quality of people’s lives and increased trust and harmony within the whole society.