I would imagine not too long after the season has ended. Hard to put a timeline on events. There are definitely strong contenders but nothing concrete as yet. I'd imagine that the board don't want to leave this decision to drag on.
I am sure the Arsenal board put out the feelers, probably before they even decided to let Wenger go. Wouldn't surprise me if the successor has already been picked, they are just waiting till the end of the season to announce it out of respect for Arsene
Over the years player transfer has been painfully slow - much easier for stellar players to leave the Arsenal and long drawn process for quality recruitment. Typically, it would occur with a day to spare before the transfer window shuts. I know there is no window for Managers leaving and joining, however, you will expect the Board to give the new guy a chance and recruit early - can partake in pre-season. Something thing tells me, they will use the World Cup as an excuse for their inertia and get someone in a few days before the start of the season! At least they cannot blame Arsene now!
It's hard to say. As the structure has now changed Minislat in charge of recruitment and Sanhelli as director of football. They have ALOT of control - which could actually put some managers off. But the recruitment process will no doubt change. There's no reason to assume things will stay the same as they were.
July 14th is the season opener against Boreham Wood, with the tour opener against Athletico in Singapore on the 16th. So you would thing that the new manager has to be in place a week or two before that. The earliest a manager who is currently working can be announced is the end of the season, Champions League final is 26th May. So I would guess the manager will be announced sometime in June.