Well there are risks involved. If the construct is destroyed all that money is gone. Constructs don't scale up after creation so at some point they are outdated as you level, as well as being expensive to create initially at the points they become available.
As for magic items, I would probably let constantly active/use activated ones work. But as the construct typically doesn't have any will of its own (it follows instructions of the creator and even that is relatively limited), the golem probably isn't capable of the "mental" aspect of activating other items. Even a command word activation requires some sort of "willing" the item to work as well as speaking the appropriate word, as part of the standard action to activate.
However, the creator could make traps with magical items or spells that the golem could trigger due to its immunities. Explosive runes on a wall, hidden behind a tapestry which the golem could rip off. Symbols of XZY as well. A Fire Trap. Contingency. There are many ways to get creative with spells and effects if you can trigger them with something that is either immune or will get a benefit from the effect.
Yep, and the OP wasn't asking about the exceptions to the rules. He asked a pretty specific question about adding stat increases (to attributes that aren't 0). Then a separate question on humanoid constructs using magic items.
When people ask questions, I try to answer the question not every possible variation of the question, as that can lead to more confusion than clarification.