DM Dubious has questions

This time, the party was better prepared. They had a plan of action. Mendez would heroically smash the door down and roll deftly into the room while Takis, standing back readies an arrow to strike whatever he sees as soon as the door is down.
Of course, for a new DM grappling with the complex combat rules, this simple action raises a number of questions.

• Naturally, Takis has a "readied action" to be triggered on the event of the door opening, and he would fire at whatever he sees. Is this right, or can you only ready an action against an opponent - e.g. "as soon as I see someone I fire at them"?¹
• If the Kobolds were ready and waiting, then why should they forgo any initiative roll, just because Takis has decided to "ready" an action?
• What should the Difficulty Class for the tumble check be?²
• What happens when fails his tumble check (which he did!)? Does this mean he falls prone, and is vulnerable, or is he actually still standing and so gets an arrow in the back of the head?
• Should the Kobolds get an attack of opportunity against the prone Mendez?³

I think the questions are difficult ones, but in the end what the DM says, goes - and I think I acted fairly. I am beginning to climb that learning curve ...

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1. This is the most difficult question. I ruled that Takis could get an arrow off with the Kobolds flat-footed, and then we went to initiative. However, I think this ruling is entirely debatable. In the end I ruled in favour of the PCs as at least they were thinking about the combat and trying to do as much as possible to prepare well for the fight instead of just saying "we charge in and kill everyone!"
2. I ruled 10 as he was also kicking down the door.
3. Yeah, he failed didn't he ;)