Properly choosing a random discard
Yes. Many of us use dice even in casual games, because we can't actually make random decisions, and it's easy.

Random means, well, random, so a nonrandom method is not following the rules. I don't think this is explicitly in the rules anywhere, though both die rolls and coin flips are mentions as methods of randomization. It's left implicit that a nonrandom method does not suffice for a card which asks for a random choice; it's not what the card said to do.


  705.3. ... Other methods of randomization may be substituted for flipping a coin as long as there are two possible outcomes of equal likelihood and all players agree to the substitution. For example, the player may roll an even-sided die and call "odds" or "evens," or roll an even-sided die and designate that "odds" means "heads" and "evens" means "tails."


It's also sometimes mentioned in rules clarifications, e.g. for Goblin Test Pilot from the Dragon's Maze FAQ:


  To choose a target at random, all possible legal targets (including creatures and players) must have an equal chance of being chosen. There are many ways to do this, including assigning each possible legal target a number and rolling a die.


But again, I don't think it ever says explicitly "nonrandom methods are not random", because that's supposed to be obvious. Any method is okay, as long as all outcomes are equally likely. Determining what methods satisfy that criterion is left up to the players, or if necessary, a judge.