You need to contact your local immigration office for details. Many don't require proof of source for international monthly transfers. However, some want to see documents proving a source of funds, such as SS documents in your home country or similar. What exactly they want is office-specific.
, depends on the time and day. You may allow 15-20 mins extra. Sometimes no queue, still takes 10 minutes. Sometimes a few people, take less than 10 minutes. Sometime 5-6 people are waiting.
Yes, the house owner is supposed to submit your TM30 as of the day of arrival to that address, especially if you are going to do any application at the local immigration office.