Do both parties need to explicitly agree upon price of service before doing it? For example, when one goes to dentist, he might ask you "Do you want me to fix cavity for you?" without explicitly disclosing the price for the service at that time. Then, if you say "Yes", they do the service and disclose price only after service is already rendered. What puts obligation to the consumer to pay for this service, if price wasn't explicitly disclosed before the service was actually rendered (e.g. when I went to the dentist for the first time, did he ask me to sign a written contract that mentions that it is my responsibility to check how much he would charge to fix cavity)? I have encountered different dentists in my life and some of them explicitly mention the price before doing the service while others don't.
The parties don't need to explicitly agree on the price beforehand, unless a law explicitly requires it for a specific kind of transaction. Contracts do not need to have all terms explicitly spelled out for those terms to be binding; it is expected that dental care has a cost, and if you don't explicitly specify the cost then it's assumed you're willing to pay a reasonable fee. They can't charge you six million dollars for a bandage, but there's a very wide range of fees considered "reasonable" for a doctor to charge. If you care more about the price, it's your responsibility to work these things out yourself before saying "yes." The dentist does not need to make you sign a written contract saying it's your job to check, because when you agreed to the care without asking about the price you already agreed to pay whatever would be charged (subject to the fact that courts will not enforce an unconscionable contract). California law requires hospitals to provide someone without insurance a written estimate of healthcare costs upon request, as well as requiring any hospital with a master list of charges for services to make it available to the public and to give anyone (on request) a list of average charges for 25 common outpatient services. It does not require medical charges to be set out in advance without a request, just like other fields of commerce aren't generally required to do that.
Not a lawyer, but: In many countries, a purchased item is your property once you removed it from the premises of the seller. In practice, this rarely makes a difference. You have entered a contract with the seller where the seller has to deliver the product, and you have to pay the money, you did your part, they have to do their part. There would be a difference if the item was stolen while in the store, or damaged by fire, or if the store went bankrupt and bailiffs took the item. If these rules apply in your country, then what they did is not theft, otherwise it would be theft (in all countries, if the store removed the door from your home after it is installed, that would be theft). You paid for a door, the store owes you a door. You have a legal contract. Both sides are bound by that legal contract. They have to do what the contract says (delivering the door that was displayed in the store), if they can't, then they have to do the nearest thing that isn't to your disadvantage, like delivering a new door. Or possible a different door that was on display. They can't just declare your contract invalid because it suits them better. That wouldn't be the case if this would put the store at an unacceptable disadvantage. For example, if thieves had broken into the store and stolen ten doors, including yours, the store might get away with returning your money. Since they intentionally sold your door again to someone else, I don't think they could use this as an excuse. I'd go once more to the store and ask them whether they want to deliver a door to you, according to your contract with the store, or if they want you to get a lawyer. A letter from a lawyer might work wonders. (Or of course the lawyer might tell you that I'm completely wrong, but they don't know that, so telling them that you will hire a lawyer might be enough).
It will vary by jurisdiction. This is a complicated area of law, but usually an advertisement or a display of goods in a shop is not an "offer" (in the contract law sense of the term), but an invitation to treat (or "invitation to bargain" in the US). The "offer" is the shopper saying "I'd like one of those please" or putting the goods on the band for the till. The "acceptance" is the checkout girl saying "that'll be ..."
Yes they can (by their contract) - Clause 3.1: The Seller reserves the right to amend prices at any time without prior notice. Errors and omissions are excepted. However, it is possible that there are consumer protection laws in your jurisdiction that prohibit this. I think this is unlikely, however.
Is It Legally Binding? While their customer service sucks, your oral authorization of the charge is legally binding (I take payments that way almost every day in my own business, it isn't an unusual business practice in small professional businesses). You authorize oral authorization of payments over the phone in the credit card agreement that your credit card company sends you every year that you don't read and throw away. The provider has to collect more information for a credit card payment over the phone than they do for an in person swipe in a credit card machine (e.g. your credit card billing address) and they are fully responsible for wrongful charges if they deal with an imposter. By regularly checking your credit card statements, you can confirm that no incorrect charges are present. Tax Issues If you want to take a tax deduction for non-reimbursed medical expenses, you simply need to tote up the amounts your are entitled to from your own records, and put it in the appropriate box on your tax forms. You don't have to attach documentation to your return. If the IRS disputes your payment, you can offer up your credit card statements and your photograph of the receipts, and if necessary, medical records to show that you received the services, to show that the payment really happened and are deductible. Your credit card company's records, reflected in your monthly credit card statements, are considered very reliable for tax purposes. You have the burden of proving that the expense was incurred and is of a type that qualifies for a deduction by a preponderance of the evidence in the event that there is a dispute that is litigated, which means that you must show that it is more likely than not that you incurred a deductible expense of that kind in that amount in that tax year. Privacy Issues While there are financial information privacy issues associated with this transaction, HIPAA, which covers medical records, normally wouldn't apply to a credit card payment that indicates the person paid, the person paying, the account, the amount and the date, but not a description of the medical services provided or to whom they were provided, which is what is normally on a credit card receipt. The financial privacy issues are also partially addressed by the provider's merchant agreement with the credit card company which contains terms requiring them to maintain certain kinds of security with respect to your financial information (which is not to say that the provider actually follows all of the requirements of their merchant agreement scrupulously, which is why data breaches happen all the time in businesses both large and small).
If you do not get anything from this "contract", and all it says is that you will pay someone a sum of money "on demand", it does not meet the basic requirements to form a contract, and in particular, there is no consideration being received by you. It would therefore not be a contract and would have zero legal force. If you do receive consideration, demonstrating that you signed the contract under duress will require specific legal advice, and you need a practicing lawyer in your jurisdiction.
It is actually because "this is important". Under US law, disclaimers must be "conspicuous" (UCC 2-316). So you can talk regularly when you're just stating the terms, but if you're disclaiming liability, YOU MUST BE CONSPICUOUS ("to exclude or modify the implied warranty of merchantability or any part of it the language must mention merchantability and in case of a writing must be conspicuous, and to exclude or modify any implied warranty of fitness the exclusion must be by a writing and conspicuous"). There are many ways to make text conspicuous, so bold or larger type would do, but all-caps is pretty bullet-proof from a technological perspective. Thanks to ohwilleke for salient citations: invalidation of a plain-type buried indemnification clause, all-caps clause held to be sufficient, law review article on the conspicuousness requirement.
You certainly can claim whatever you want. Will you be granted it though? Time is money indeed, and replacing the damaged items requires some time not just items themselves. The question is how much time and at what rate? Your duty will be to mitigate the damages and so to minimise the time and its cost. Will your normal hourly rate be applicable? Only if you have to use your actual work time and miss out on earning for those hours. It will probably be unreasonable to use work time in the first place, let alone that you may not necessarily miss out any earnings in case it's a salaried job and your boss is fine to just let you go for a few hours to sort out personal issues. If the time you use for replacement is evenings/weekends then you would need to quantify how much you miss out on not doing what you would do otherwise.