Which is utter nonsense. Lots of things happen in a turn, and if it took a full round for the spell to execute, it would take effect at the start of your next turn. And you wouldn't need a saving throw since you could just step out of the object's path. The catapulted object travels 90 feet in exactly the same time a lightning bolt travels 100 feet.
Your turn covers 6 seconds because despite the initiative order they all happen at the same time. Also I included above the actual procedure if you want to assume it takes less than 6 seconds. You really have two choices here - either it obeys the laws of physics and is traveling relatively slow (because it only goes 90 feet before stopping) or it is not obeying the laws of physics in which case the rest of the argument is irrelevant.
A lightning bolt does not have momentum (because it has no mass), so the physics governing its travel are different and to be clear an actual lightning bolt would not stop until it reached ground, so again here if it stops in mid air at 100 feet it is not obeying the laws of physics .... which is ok because it is magic!
Finally I will add that lightning on earth travels at about 270,000 mph. A Handkerchief traveling at 270,000 mph would obliterate you if it hit you. If your argument is now that the catapulted object is as fast as a lightning bolt travels in real life then your "open" net would liquify anyone it hit. It would sink a battleship, punching through its armor with ease. To understand the comparison here, a railgun will shoot a projectile at about 6000 mph, lightning is over 40 times faster!
The reason it is possible to survive a lightning strike IRL is explicitly because it has no mass and no momentum.
DM's call. But if if you did catapult a blanket it would ONLY do damage, it wouldn't wrap the person up.
Everything is technically the DMs call, but if he rules no damage he is overiding RAW. I agree a blanket does not wrap someone up because while a blanket is on the table on page 150 of the PHB, but it is not further detailed in writing on page 148-152 meaning it has no "special rules"
as noted on page 148.
A net on the other hand does have additional properties detailed on page 148. As DM you are free to overrule those or any rule in the game or to rule they just don't apply in this situation. I just don't know why you would.
No. They do not. They say if you make a weapon attack using a net and the attack hits then the target is restrained.
That is not what it actually says, it does not mention a weapon attack at all. This is entire text on on page 148 of the PHB regarding a net -
"A large or smaller creature hit by a net is restrained until freed. A net has no effect on creatures that are formless, or creatures that are huge or larger. A creature can use its action to escape from a net with a DC10 strength check, freeing itself or another creature within its reach on a success. Dealing 5 slashing damage to the net (AC10) also frees the creature without harming it, ending the effect and destroying the net. When you use an action, bonus action or reaction to attack with a net you can make only one attack regardless of the number of attacks you can normally make"
There is nothing at all that mentions a weapon attack with respect to the restrained condition and other items detailed in the PHB do make that distinction when it applies (seeLance, Acid, Alchemists Fire, Oil and Holy Water). Further when they mention an attack later in the text they again do not even specify the type of the attack as a weapon attack, only mentioning an action, bonus action or reaction that allows you to attack. This is notably different than other items, including weapons, where they do specify specify "attack"
to achieve the desired effect.
I will also add that a published WOTC adventure actually has a net that drops on adventurers and restrains them without an attack being made (Ghosts of Saltmarsh, Chapter 6, Fortress Level 1, Room 1).
I think you are navigating this with feel instead of reading the rules on it as when you originally questioned it you mentioned grappled, where net actually causes restrained.