On point one, CPI uses a combination of "owners equivalent rents" and more traditional rent measures.
Owner's equivalent rent is basically just a survey where they ask homeowners how much they could rent their house for. So survey participants understanding of market rents may lag.
But more importantly, when they survey renters, they ask them what they're currently paying. So if somebody is in a one year lease, and gets surveyed in month 11, they will give a rent figure that's almost a year old. On top of this, CPI includes rent controlled units, below market rent units etc. In some sense this is "correct" because it reflects what people are paying... But the whole value of CPI is to be a forecasting tool. Using lagging metrics is bad design imo.
Finally, they only survey 1/6 of the housing stock each month. So the whole sample is not updated every month.
To your second question, the biggest change is that CPI used to use home prices rather than the OER measure. That metric would show 15-20% rather than the 5% we get from their current formula. Given that the shelter component is the biggest weight in the CPI, it would shift the number up a few points.