How do I solve problems that occurred in a new building shortly after it was handed over?

First, as an architect and technician, I regret that the end users unfortunately have to suffer the consequences of poor workmanship. Unfortunately, there are many agents involved, and often those who enter, if they are not good professionals, do their job and leave without caring whether it is better or worse done, and it is almost impossible for project management to notice many of these failures until the end user starts using the house.

As for the problems you are having, I will give you some recommendations that may help you solve them until you clarify who is picking up the costs, as in your situation I would fix it myself so I would not live in these conditions, I would keep all receipts and bills and then present them along with the complaint to the builder, as they should be the first to give you a solution, even if they then shift responsibility to a third party:

— The problem with the flashing, since you are explaining what is happening to you, I assume it is poorly sealed, unfortunately this is a problem that happens all too often, the solution in this case would be to remove the flashing and seal all the holes with foam. If they are well sealed when the flashing is removed, it could be due to a poorly sealed shutter box (you can also use a thermal shutter box, which is well insulated and improves heat and sound insulation), or poor quality joinery that is not double glazed or thermal break (although this is almost impossible if it is new construction).

— Smell in bathrooms is a more delicate issue because it's not always easy to determine where it's coming from. It is usually caused by poor sealing of the siphon canister or toilet bowl, so only repairs will be needed. However, it can also be due to other causes, for example, if your home is on the top floor, the vents may be too short, so they need to be lengthened so they come out above the draft vents on the roof floor. Another reason could be poor ventilation, although this is more difficult because the user usually installs the vents, sometimes the clock is included in the electrical panels as a control mechanism for the mechanical exhaust (as I suspect the bathroom is indoors), and sometimes the developer does not explain their use to the end user, so sometimes they may not be set directly or at very short intervals of ventilation.

— The third point is by far the most intractable, because from what I understand you are saying, the access to the house is through a hallway that is completely or partially unenclosed, so that when the wind blows water gets in. The problem may be that the slopes of that hallway are poorly made, so the water is not draining properly from all points, this can be checked by throwing a cauldron of water from your door and seeing if all the water is headed to the drain or downspout location. Another reason could be clogged drains, if any, or too few drainage elements (gutters or downspouts) for the area they are supposed to drain, so when it rains they can't drain water fast enough, causing water to accumulate and enter homes. With the exception of the drain cleaning issue, other problems are not as easy to solve and will need to be evaluated on site to see what measures should be taken.

— Finally, the last of your problems seems to be a consequence of the previous one, because if water gets into the house, I imagine it also wets the door, so it can start to expand at the bottom (which you can see with the naked eye), which will cause it to rub against the floor when opening and closing, making it difficult to use, plus water rises due to capillarity, so this moisture can rise along the door over time. It can also expand during very sudden changes in temperature, so this will probably be more relevant in the winter than in the summer. To solve these problems, you can make the door flat a few millimeters from the bottom to provide more clearance from the floor. Finally, although less likely, the hinge may come loose and not be adjusted properly and the door may nod off a bit, in which case this can be corrected by adjusting the hinge.