Between 2011 and December 2017, I made online purchases from Apple Store Malaysia online: an 27" iMac, 3 MacBook Pro's, 2 iPads, 2 iPhones and 1 iPod Classic. All delivered with no issues and all still in use today. While I was studying, I applied for education discounts, no proof was asked of me. For the first Mac I purchased, I did receive a phone call from Apple Singapore; I thought they wanted proof of education, I asked where to send it. However, lady said, not necessary, she only wanted to verify address and credit card details as it was first large sum purchase from my Apple account online. All MacBooks and iMac, education pricing were applied for, and proof was never asked from me, but I guess they could do a random check in the future. Delivery was done by DHL Express, all were trackable. Main reason why I ordered online was because I wanted them spec'ed up, and Machines (and other Mac brick & mortar stores) often do not carry different models other than the base/standard models. Also, engraving was done by the online store too! I got my iPod Classic engraved. 

To share another positive experience, a friend of mine purchased iMacs from Apple Store Malaysia online for work; when he had issue with one of them, he called Apple customer support and had a conversation with Apple on the phone (which included trouble-shooting and then agreement that some thing wasn't right). Arrangements were made with little hassle. The guy providing customer support sounded like he was based in the USA, and was very friendly, there was no fuss to get it returned with no need of documents or proof. He just said, put the iMac in the box it came with. Apple had DHL Express picked it up right from his house to be shipped back to Apple for free - whilst under warranty of course. Saved him a trip from carrying it all the way to a store.

You should not have any problem servicing your apple product at the authorized service centers if you buy online. Usually the apple warranty online should be valid at their authorized service providers also. But if you jailbreak your phone or made modifications to it on your own then won't apply already. If you're having problems you can email apple then bring the warranty details or email from apple as proof.

Yeah, I had no problems; walked into Machines at Mid Valley before to have my MacBook Air serviced. They don’t care where you purchase it from, just make an appointment, and then you speak to their service team, and they will try to trouble-shoot. The costs of service or replacing any parts will depend on whether your device is still under warranty. They will send your device off to Apple Singapore or their local authorised service centre. You will just have to bear any costs incurred, if out of warranty. If still under warranty, or if you purchase Apple Care, any Apple Store globally will accept servicing for free. I had another issue with my MacBook Air purchased from Apple Malaysia online store, and I went to the UK for studies, and Apple UK fixed it for me free. No proof of purchased needed because you would have purchased your Apple device using your Apple ID, and that is linked to the Apple Store online and has all of your purchases made in history, which has serial numbers. So they have warranties attached to those devices automatically.