The turn structure looks like the following:
1. Trade/build. All players may trade properties, mortgage/unmortgage properties, and buy/sell houses/hotels. If the current player is in Jail, they may pay $50 or a get out of jail free card during this step.
A. Roll and move token. If you land on or pass go, collect $200. If this was your third doubles, instead go to Jail and and the turn. If you are in jail and your roll does not get you out of jail, end the turn. If you are in jail and it is your third roll, pay $50 and move your token as normal.
B. Resolve the space you landed on. If it's unowned, purchase/auction property. If it's owned by someone else, pay rent. If it's chance/community chest, draw and resolve a card. If it's go to jail/luxury tax/income tax, follow the instructions on the space. While resolving the space you landed on, players may trade/build as in step 1, but only after the effect of the space has been determined.
3. Handle doubles. If you rolled doubles, go to step 1.
The support for this comes from both the Monopoly Millennium Edition rules and the current standard Monopoly rules.
Building and mortgaging can be done at any time:
You may buy and erect at any time as many houses as your judgement and financial standing will allow.
Houses and hotels may be sold back to the Bank at any time
Unimproved properties can be mortgaged through the Bank at any time.
Trading properties is not mentioned as something that can be done at any time, but it is also not listed with a timing restriction.
"At any time" means that you can do it on anyone's turn, while paying rent, etc. However, rolling, moving, and the determining the consequences of the space you land on are atomic:
[on your turn] throw the two dice and move your token
When you land on a property owned by another player, the owner collects rent from you
Consequently, another player may not build properties after seeing what you have rolled but before determining what rent is owed.