All right here are a bunch of questions and some help would be appreciated. Some are easy but I'm just not too sure.
I'm going to assume high school rules here
-----For girls high school is the five second count only in the frontcourt and only after the dribble has ended? Holding the ball?
Same as for boys. 5 second count is only in the front court, and is while either holding or dribbling, though a new count is given each time the player goes from holding to dribbling and vice versa
1.Does the arrow switch- halftime? If Team B has the arrow at the end of first half does that mean they will start with the ball in the second half? If Team A gets the jump to start the game does that mean Team B gets it in the second even though the arrow was pointing to team A to end the first half?
The arrow points to the basket of the team that will receive the next AP throw-in. So yes, they switch the arrow at half time because the teams switch baskets.
2.When a player saves a ball does he/she have to get two feet in? Or one foot other not touching out? Can they dribble after saving it? If A1 is dribbling fumbles the ball then he can’t dribble again? If he saves the ball after it fumbles then I assume he can’t dribble it. If A1 catches a pass and it goes off his hands and then he saves it can he dribble the ball?
After saving the ball, if the "saver" has something touching in and nothing touching out, he may be the first to touch the ball. Unless he has previously dribbled, the saver can dribble the ball. As for a fumble, a player may fumble-dribble-fumble, but not dribble-fumble-dribble.
3.If A1 goes for a layup gets blocked and then fouled by a second player or the same player say on the landing- when does the shot end—would it be on the ground?
A foul on an airborne shooter is considered a foul on the shot. However, if the shot is blocked then the player is fouled, there's a better than average chance that I won't call the foul. It really depends on what kind of contact it is.
4.I’m not totally clear on 3 seconds.
Guy goes in the lane for two seconds- get passes the ball holds it for 3 or 4 seconds then passes it? Is that 3 seconds,, if not should he get out of the lane right away- or would the ref count to 3 before making the call? If he was in the lane for two seconds, gets the ball, two seconds more and then passes it out is that a 3 second count if he doesn’t get out of the lane within a new 3 second count.
No self-respecting official ever calls 3 seconds, unless forced to. If people in the stands are yelling for it, it almostly certainly isn't. People in the stands who holler for three seconds are mentally incompetent and were not loved as children. Technically no player may be in the paint for 3 seconds while his team is in possession of the ball in its frontcourt. Allowance is made for a player who is making a move on the basket. Do a search on 3 seconds for more details.
5.Airball- shot- can he dribble it? Does it matter if it hit the ground?
On any shot team control ends and any player may get the rebound and treat it like a new possession. If the official thinks the airball was a shot, it was.
6.On the free throws the players have to wait til the ball hits the rim? Not the backboard.
What if all it gets is net? Do they have to stand their forever? 
Rim or backboard, whichever comes first.
6.B—Say at the end of the game, a free throw misses bounces away without touching anyone. The timer started the clock. Time runs out. If this is a timer mistake then the ref can put time back on the clock but how do you know who gets the ball. I assume it alternates but if team A the free throw shooting team gets the ball I could see that team B would be upset. If Team B is clearly going to get the rebound or gets the ball and then time runs out would the ball go to Team B with time on the clock? OR at the end of the half a team rolls the ball in so the clock doesn't start, but the timer starts it too early. If the ref doesn't see this and a shot is made but after the buzzer is it waived off even if without a doubt the shot would have been legal if the clock started on time. I would assume the play wouldn't count.
6.B? Are you ****tin me? In my day 7 came after 6. If the ball becomes dead with no team is in possession, play is restarted with an AP throw-in. Who cares if B is upset? Yes, the offical can put the time back on the clock if he has definite knowledge of how much time should be on the clock. Do a search on this and you'll find many threads with fascinating and contentious discussions on this point.
7 510 1E in the Case book- Does clock always run, after a made basket? I thought it stopped. Does it run once a player has the ball and is ready to throw it again? Or is this just late in the game.
In HS the clock does not ever stop on a made basket.
8- Backcourt- Again not too clear. Mainly I want to know when the frontcourt is established. Can A1 with both feet in the backcourt dribble in the frontcourt and then back to the backcourt?
Can A1 step with one foot or both feet in the frontcourt but the ball dribbling in the backcourt? And then return the feet back to the backcourt?
I saw a player run along the midcourt- one foot in the backcourt the other in the front court dribbling the ball in the backcourt and then the frontcourt and back again. So the ball and one foot was at times in the front court but not both feet.
Until both feet and the ball are entirely in frontcourt, a player who is dribbling is still in backcourt. He may cross the midcourt line into front court with one or two points as many times as he wishes, until he has all three points in front court. After that, he may not touch on or over the line with any of the three points.
If a train leaves Boston and 10 am traveling 45 mph and Train B leaves Houston ---...
Both trains will derail.
---Is it walking if a player who is on the ground after a loose ball gets to one knee holding the ball? Or if on the ground tries to dribble it and stands or gets to one knee?
A player may not travel while dribbling. So starting a dribble then getting up is legal. If a player who gains possession of the ball while laying on the floor attempts to get up without starting a dribble, it is traveling.
After a 3 seconds call where is the location for the inbounds throw? For fouls or violations not on the baseline is the inbounds throw always freethrow line extended or above? If it is a charge near the basket does it go on the side or baseline? I heard something about not having throwins in front of a visiting teams bench. If the ball goes out there do you just throw it in there?