Though you can of course play this however you want, I don't really see any support in the rules for this interpretation.
In the ambush scenario, whatever benefits the ambushers may accrue come from their enemies being surprised which is determined via their Dexterity (Stealth) check against the PCs' passive Perception scores. The PC with the Alert feat is simply immune to having this condition placed upon him or her while he or she is conscious. First surprise is determined, then positioning, then every participant rolls initiative. Once in initiative, the ambushers can take the Ready action, but at that point, the PCs are also in Initiative, even if they can't see or hear their assailants yet. They can't Ready before Initiative.
For the rug, it's resolved the same way: The Dexterity (Stealth) check of the rug is compared to the PCs' passive Perception scores. Anyone whose score it beats is surprised (except for PCs with the Alert feat) and cannot act on their first turn and can't take reactions until their turn.[/QUOTE
No, but its the only reasonable interpretation. If you do it as you say, the players themselves wouldn't be able to set an ambush. More importantly, the simple scenario of "I ready an action to fireball the room as soon as the fighter opens the door becomes impossible, as according to you the combat would not start until the door is opened therefore the wizard could not ready an action." You cant tip you players off to an ambush by saying roll for initiative, that's just dumb. Think about the situation as whole within the framework of the rules, don't try to fit the encounter into the rule set word by word.
For the Rug it specifically says that it is indistinguishable from a normal rug until it moves. It is an actual object, a rug, until it animates. A creature that has advantage on its perception checks due to sight or hearing would just see a rug, a creature with blindsight or tremor sense or anything like that would just see a rug also. The description specifically says those able to detect magic though will see it. That means to me that no player will be able to even try a perception check unless they say they are making sure the rug isn't something else, PC don't get an auto-detect system superior to anything in the game except for detect magic. Otherwise the rug is no challenge and as written in the Monster Manual description
Would-be thieves and careless heroes arrive at the doorsteps of an enemy's abode, eyes and ears alert for traps, only to end their quest prematurely as the rugs beneath their feet animate and smother them to death. A rug of smothering can be made in many different forms, from a finely woven carpet fit for a queen to a coarse mat in a peasant's hovel. Creatures with the ability to sense magic detect the rug's false magical aura.
would almost never occur with any group of reasonable adventurers as somebody would beat the Rugs check (+2 modifier) well before they would get within range of its attack. However, PC who use detect magic get the benefit.
A Human wisdom focused PC with 16 Wis and Observant feat has a passive perception of 18 at level 1, you cant let the group just him walk around the world using his spider sense as an early warning system. If that was true every single NPC group and enemy group in the game would do exactly the same thing, and barring the occasional group using mass invisibility with silence the PC wont be able to spring surprises on the bad guy.
If you watch any of the D&D plays on Twitch like Stream of Annihilation this is effectively what they are doing and how they play it.