Can my brother in law be in the house with me during the day or does the door have to be left open?
Additionally at night when my 9 year old son and 8 year old daughter are home what would be the halacha of having him in the house when my husband is out. May I lock the door?
The door must be left open when you are alone with brother in law.
At night your 2 children would provide shemira to prevent a yichud problem. However, you must also determine that the situation does not violate the guidelines of tznius. So if this is a one time thing that may be ok, but on a regular basis it would seem questionable if such a situation is proper behavior.