Indeed welcome to the forum. If you're new and preparing for your first rules exam, I think you'll find this forum to be a great study aide.
IRT your question, I would note that this is why most officials agree that Rule 4 (Definitions) is the bedrock for all other rules understanding.
See Rule 4-4-7d: A ball is at the disposal of a player when it is:
.....d. Available to a player after a goal AND (my emphasis) the official begins the throw-in count.
What is "available" is up to the new trail's judgment. Generally accepted practice is that, if the non-scoring team is slow to react after the score, we allow them a little time. If they pick up the ball, start counting. If they don't, and the ball is available and they're just sluggish, start counting only when it gets to the point of ridiculous*. Of course if the ball gets stuck under a bleacher or lost down a hallway, blow the clock dead in order to resolve that.
(* Maybe a little sooner at the end of a close game where the non-scoring team may intentionally be trying to waste time.)
Now, you asked about disposal WRT granting time outs. Most coaches who want to call time out after a score know that they have to do this immediately. Understand game situations and be ready for this. What I say next is unwritten but solid advice: if said coach is asking for the time out immediately after the score, but the non-scoring team is already gathering the ball and taking it out by the time you confirm who is calling the time-out and/or put air in the whistle, grant the time out. In other words, if the intent to call the time out while the ball was dead is clear, give that coach/player the benefit of the doubt.
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