Learn English – What does “kind of sums things up” mean

Various dictionaries have different things to say.

What price [fame/success/victory etc.]?

something that you say which means it is possible that the fame, success etc. that has been achieved was not worth all the suffering it has caused

What price victory when so many people have died to make it possible?

(Cambridge Idioms Dictionary, 2nd ed.)

price [...]

what price (something)? what are the chances of (something) happening now?

(Collins English Dictionary – Complete and Unabridged)

What price something?

What is the value of something?; What good is something? (Said when the value of the thing referred to is being diminished or ignored.)

Jane's best friend told us all about Jane's personal problems. What price friendship? Jack simply declared himself president of the political society. What price democracy?

(McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)

what price ——? used to ask what has become of something or to suggest that something has or would become worthless : what price justice if he were allowed to go free?

(New Oxford American Dictionary 2nd edition, from OS X)

Apart from Collins's strange definition, it seems that the general meaning of "what price X?" is "what's the value of X?" (not "what's the price of X", in the modern sense of price).

As for how it came to be, my wild speculation of the day is that it could have been used grammatically in an anaphoric expression, eg What price is freedom to us if we tolerate this tyranny? What price justice? etc.

It's usually meant literally, but "tea" doesn't just refer to the drink, but refers to the meal known as tea. There are quote a few different types of meal called tea, depending on which country or which part of the UK you come from.

For example, there's "low tea", or "afternoon tea", usually a light snack such as sandwiches or eaten between 2pm and 5pm. Nowadays, this can also be a treat in a cafe or hotel with cakes, pastries, or scones and jam and cream. (See this book and this book.)

There's also "high tea", an early evening meal, eaten between 5pm and 7pm. This would be followed by a larger meal later on. Low and high refer to the height of the table where tea was eaten.

Tea can also refer to the main evening meal in the north of England, Scotland, Wales and Northern Ireland.

"Taking tea" is a common phrase for having this meal or snack or just sitting down to brew up, pour and drink tea; it is somewhat of a ceremony. For what it's worth, here's an Ngrams chart (and there's not much difference in shape between UK and US English):

"Children to compete for chance to take tea with the Mad Hatter."

This clearly refers to the Lewis Carroll's Alice's Adventures in Wonderland where in chapter 7 the Mad Hatter is having a tea party. It's always six o'clock, and they're always drinking tea. There's bread and butter laid out as well:

'Take some more tea,' the March Hare said to Alice, very earnestly.

'I've had nothing yet,' Alice replied in an offended tone, 'so I can't take more.'

'You mean you can't take less,' said the Hatter: 'it's very easy to take more than nothing.'

'Nobody asked your opinion,' said Alice.

'Who's making personal remarks now?' the Hatter asked triumphantly.

Alice did not quite know what to say to this: so she helped herself to some tea and bread-and-butter, and then turned to the Dormouse, and repeated her question. 'Why did they live at the bottom of a well?'

But as to the original question, the winning children would join the Hatter's tea party. There might be actual drinks and snacks for the children, or it could be meant that the winners would get to spend time with the Hatter.