“Max speed — [ability]” means “As long as you have max speed, this object has [ability].” If the granted ability functions in a zone other than the battlefield, the max speed ability does too.

If an effect needs to know what a player’s speed is and that player doesn’t have a speed, their speed is considered 0.

The copy remembers any decisions that were made for the original spell as it was cast, including values chosen for X in its mana cost and whether any alternative or additional costs were chosen. For example, if the original artifact spell had kicker and its kicker cost was paid, the copy will also be kicked.

A resolving copy of a permanent spell becomes a token, so the token isn’t “created.” Effects that care about a token being created won’t interact with a token that enters the battlefield from Mendicant Core’s last ability.

A player “has max speed” if their speed is 4.

Start your engines! isn’t a triggered ability. Increasing your speed to 1 is something that happens as a state-based action as soon as you control a permanent with the ability. Notably, this includes gaining control of a permanent with the ability that another player controls.

Your speed doesn’t change until a spell or ability says so, such as the inherent triggered ability that cares about opponents losing life during your turn. Notably, losing control of permanents with start your engines! doesn’t affect your speed.

Each player tracks their speed (or lack thereof) separately. Increasing your speed has no effect on whether another player has speed.