You can apply for the DTV, after you get your Visa number then you can apply for your son, citing your visa number as the primary holder, pay the fee for the dependant, and provide a financial sponsorship letter and your funding information again.
There's no requirement that you need to have accommodation but it doesn't hurt to have a booking for a week or so while you look around for something.
You can bounce out at 180 days and come right back in 5 minutes.
If you're applying as some kind of remote worker or a freelancer you will probably need to justify it with bank statement showing pay slips, that kind of thing. You can provide all that in the beginning or you can wait for them to ask for it, if they do. You simply need to convince them that you are a genuine freelance worker and have a client history and possibly a portfolio.
Completely depends on which embassy you are applying at. Some want to see the funds there for 3 months, others don't care if it was just put in the day before.
Congrats on the nature of the letter, and sticking to your guns. Please explain exactly how you convincingly approached your treatment plan in the letter as being better suited to the DTV.
all visa applications are online now, no embassy visit unless you are called in for an interview. You choose the particular embassy to apply online according to where you are physically located.
Initially a 6 month course will suffice. If you border bounce at 180 days you won't need to prove ongoing activity, but if you do an in country extension you will need to update and prove basically all the initial application criteria, plus some.