If I equip Blade of the Ruined King to one of my units, do I kill another unit as part of starting a chain? Can the chain be reacted to and, for example, my opponent play Star Crossed to bounce the unit I want to kill to hand? Or if they bounce the unit I am equipping it to, does it stop the chain and/or stop me from killing my own unit?

To address your questions regarding Blade of the Ruined King, here is the breakdown of how the mechanics and timing work:

1. Does killing a unit for the Equip cost start a chain?

Yes. Activating the [Equip] ability of Blade of the Ruined King starts a chain.

According to the rules, activated abilities are placed on the chain as a pending item before they are resolved. Because the ability is now on the chain, it creates a window for players to play [Reaction] cards.

2. Can my opponent play Star-Crossed in response?

Yes. Once you activate the [Equip] ability and place it on the chain, your opponent receives priority and can play a [Reaction] spell like Star-Crossed.

• Sequence:
  1. You activate the [Equip] ability. You pay the [C] cost and kill the target unit as the cost, then the ability is placed on the chain.
  2. Your opponent plays Star-Crossed in response to your activation.
  3. Because of Last-In, First-Out (LIFO) resolution (Rule 340.1), Star-Crossed resolves first.
  4. Star-Crossed returns your targeted unit (the one you intended to equip) to your hand.

3. What happens if the unit I am equipping is bounced?

If your unit is bounced to your hand by Star-Crossed before your [Equip] ability resolves, the following occurs:

• Ability Whiff (Mistargeting): Your [Equip] ability remains on the chain targeting the now-absent unit. When the game attempts to resolve the [Equip] ability, it checks the legality of the target. Because the unit is no longer on the board, it is an illegal target. Consequently, the ability will fail to execute its instructions (Rule 359.3.e.7).
• Cost Payment: The [C] energy and the sacrificed unit were already committed during the activation process; they are not refunded or returned.

4. What if they bounce the unit I intended to kill?

Because the unit is killed as a cost during the activation process (before the ability is placed on the chain and before the opponent receives priority), it is impossible for an opponent to bounce that unit in response to the activation of the [Equip] ability.

Summary: You cannot force the kill if the unit is no longer there, but because costs are paid upon activation, the sacrificed unit is removed from the board before your opponent can respond to the [Equip] ability.

Note: I could not fully verify this ruling against the retrieved rules.