Since you can double-spend your old Bitcoin coins and the Bitcon Cash coins, independently---one on each side of the fork---it seems somehow that money was created for free. Isn't this a problem? What prevents people from creating more and more forks to the point of undermining the crypto-currencies value?
To answer the questions in the body of your posting:
Spending BTC "on one side of the fork" and BCH on the other cannot be called "double spending". "Money created for free" is a misnomer. Yes, you have twice as many coins after a fork, but the combined value of both coins "right and left of the blockchain" is initially the same as just before the split, namely the coin with the former name has 100% and the one with the new ticker 0%. But from that point on price discovery on the market (exchanges) will determine the value of each strand individually.
"What prevents people from creating forks . . .?" Among others the risk of loosing trust and competitiveness in regard to other crypto currencies. If nevertheless a fork is triggered by some actors in a respective community, there must be important reasons why it couldn't be avoided. A fork is a declaration of war, after all efforts to compromise and reaching consensus failed. But as we can see in case of Bitcoin, a fork does not necessarily undermine the value of that currency. The combined value of Bitcoin and Bitcoin Cash is now, 3 weeks after the fork, considerably higher than before.
Yes, it is created out of thin air. Even I can create a fork of bitcoin anytime and announce it into the market. The huge difference is probably it won't be valued as much as BCH.
Isn't this a problem?
No, it is rather advertised as a feature of bitcoin. Users, if they don't like anything about the current bitcoin rules can always fork away.
What prevents people from creating more and more forks to the point of undermining the crypto-currencies value?
There is nothing preventing forks from occurring. The important part of every fork to survive is ecosystem support. This roughly includes merchants, bitcoin services/businesses, miners, exchanges and users.
Along the same lines, bitcoin is also created out of thin air. It is just that market believes btc to be worth something.
In a zero sum game, one would expect that the sum of the tokens would retain approximately the same total value.
However, the spin-off of Bitcoin Cash reduced the uncertainty for each of the irreconcilable visions for Bitcoin, each now having a project aiming for their respective priorities. Thus, it seems plausible that investors waiting on the sidelines became willing to enter the market.
Personally, I suspect that a lot of BCH was bought with FIAT, and the sellers of BCH mostly invested it back into Bitcoin. Some additional investments into BTC with the pending activation of SegWit drove both coins to new heights.
(After getting 4 downvotes and a comment, I complied by removing the greater part of this answer, because it was considered polemic. Sorry for having been ignorant about the purpose of this list as Q&A site.)
A fork of a blockchain can be compared to splitting the shares of a company, e.g. by increasing the total no. of shares from 100 to 200.
The value of the company stays the same, but instead of 1 share, you now have 2, but they still only represent 1% of the value of the company. The value of a company and its shares is determined at the stock exchange by monitoring at what price the shares are traded. Initially the 2 new shares will have about the same value as 1 old share. But that changes, depending on how the future development of the company is anticipated.
After the fork the prices of BTC, which has kept the legacy name, and BCH are determined separately on the exchanges.
I don't think that this process can be called "creating money from nothing". Additonal tokens have been created and given value by people who are buying it for a reason.
When a government or central bank "prints" money it kind of taxes the people by diluting the value of the currency.
Crypto miners get coins as a reward to process transactions and maintaining the systemic process. The blockchain protocol is designed to release only relatively small amounts of new coins on a regular basis to not cause "inflation". Still the holders of coins are "taxed" that way, but it is a fair price to pay for being able to use a currency without "strings attached".