At this point the following rules come into force:
3-3-6 . . .
A player who is bleeding, has an open wound, has any amount of blood on his/her uniform, or
has blood on his/her person,
shall be directed to leave the game until the situation is corrected, unless a time-out is requested by, and granted to, his/her team and the situation can be corrected by the end of the time-out.
10-5-3 . . . The head coach must replace or remove a disqualified/injured player, or
player directed to leave the game, within 30 seconds when a substitute is available, while within the confines of his/her bench.
2-12-5 . . . Sound a warning signal at 20 seconds of
the 30 seconds (maximum) permitted for replacing a disqualified or injured player, or
for a player directed to leave the game.
And most likely this one too, despite the last part which would only apply if there was a player DQ:
2-8-4 . . . Notify the coach and
request the timer to begin the replacement interval, and then notify the player on a disqualification.
What exactly is this 30 seconds?
It is a replacement interval, not a time-out.
In 2003-04 the NFHS addressed this in a POE. POE #5 that year was called "Substituting – Player Disqualifications," and uses language specific to disqualified players or players who have "fouled out",
but it can also logically be applied to any case in which a player is directed to leave the game and must be replaced in such time. That is the case here.
What warning horn are you talking about? The one from the original time-out or the one from the 30-second replacement interval for the bleeding player? You will see below that either horn doesn't matter in this situation.
Did you instruct the timer to start a 30-second interval following the second horn from the time-out? Since you are asking if you did this right, I have to tell you that you didn't, if you failed to properly invoke the 30-second interval.
That is a good point about which to inquire. Since your situation did not involve FTs at all, I would say that the exception for 3-3-1c doesn't apply and that you need to focus on the rules pertaining to the 30-second replacement interval.
From the 2003-04 POE those are:
H.If a substitute from the non-offending team or a substitute from the offending team for someone other than the disqualified player reports to the scorer during the wait for the required substitute, they may enter the game. (3-3-1d) (Notice that the rule citation given is not the FT exception.)
I.A warning horn shall now be sounded at 20 seconds if the required substitute has not reported to the scorer’s table. This is an indication to the head coach that a substitute should report immediately (2-12-5). This warning horn does not have the same substitution restrictions as a warning horn for a time-out (3-3-1a); other substitutes from either team may still report and enter the game until the ball is about to become live. (3-3-2)
I don't agree with you there. The player with blood on him NEVER left the game, so he doesn't have to reenter at this point. He is already "actually entered." He is merely required by 10-1-9 to return to the floor at approximately the same time as the other four players. You said that he was on the court in front of the bench with his coach looking at his elbow, so he met that requirement.
How can you say that the sub is not a "sub"? 
Was he legally in the game when the time-out was granted? No. Was he legally in the game when the warning horn sounded during the time-out? No.
He is simply the required substitute for a bleeding player who was directed to leave the game following the time-out.
That's my opinion on how the rules should be applied in this case. I truly do hope that my writing it out helps you in some way. 
Of course, if you don't agree with it, then just ignore my post.