In the same vein, are you allowed to run a proxy (a real card, like an overwritten basic Island) at sanctioned tournaments, if you have the real thing in your deck box?

No.

3.4 Proxy Cards

A proxy card is used during competition to represent an Authorized Game Card that has been accidentally damaged or excessively worn in the current tournament (including damaged or misprinted Limited product) as determined solely by the Head Judge. Proxies are not allowed as substitutes for cards that their owner has damaged intentionally or through negligence.

Players may not create their own proxies; they may only be created by the Head Judge. When a judge creates a proxy, it is included in the player’s deck and must be denoted as a proxy in a clear and conspicuous manner. The original card is kept nearby during the match and replaces the proxy while in a public zone as long as it is recognizable. A proxy is valid only for the duration of the tournament in which it was originally issued.

Official checklist cards are Authorized Game Cards and may have a proxy issued by a judge.

How about stores running "rent-a-deck" deals?

How timely! I had this idea about 2 weeks ago; like 99% certain that it was not original. And in fact there is an online service that you can rent a constructed deck from. But by mail seems really dicey; the prices are like 15% of list -- so you are talking 50-100s to rent a deck. That doesn't seem like it has a market.

One of the whole cool ideas of renting, say Standard decks is that you can switch whenever you like... so it seems like even $50 is kinda high.

I think people would be willing to pay maybe $20 to rent a tier-1 deck for a $10-20 entry constructed tournament. And I think to keep the price that low, you would have to rent the cards from the store in person, and I am not sure the store makes any money on this (given attrition to wear/tear).

Are there paper stores that do this? How much do they charge?