I’m no economist, but it seems like the millions of dollars of electricity that have gone into mining crypto currencies is transferred into the underlying value of the crypto currency. So if I know that it will take $100m of today’s money to mine crypto currency for the next year, then I would expect the total market cap of that crypto currency to rise by at least $100m with a premium for speculation and other supply demand issues. So in that sense the value of the mined currency should increase with the mining cost.

It's the other way around really, the value of the crypto dictates how much money you can spend mining it and still break even. Except then the difficulty will increase and even things out. Having more miners could increase the trust in the currency and increase its value but it's not directly correlated.

Mining from a random browser will almost certainly never be cost effective for the user, you're better off buying or renting dedicated hardware, mine on it and use the reward to pay for microtransactions.

But that is not the right question to answer. The right question is is the cost of mining crypto currencies in a browser more or less than the cost of dealing with advertising?