Can a soldier get promoted on his promotion date and get back pay after being medically flagged for 8 months?

You cannot be retroactively promoted unless the flag was erroneous. From reading your posts you were flagged for ht/wt, but have a medical condition. AR 600-9, para 3-11 defines how a provider could issue a temporary extension in ABCP, but it is very rare to qualify for this. No medical condition will waive your flag and you have to return to standard to overcome the flag. If your medical condition is truly that bad the extension will give your docs time to work and try to control it. Once the extensions end, max 12 months, you must meet standard or you'll be chaptered in time. If that becomes the case maybe you can be medically separated if the medical condition is bad enough. Otherwise, I wish you the best in your care and all of this is outlined in AR 600-9.

No a Soldier starts receiving Pay increase on the effective day of the orders.

Yes, talk to your HHC (or whatever you call your Headquarters CO) MEDO. If the flag ended up being erroneous, due to a medical condition (say, them failing a tape test was proven to be genetics--or something like that), then the flag would be removed and basically, any record of the period they were flagged would be scrubbed from their record/ERB.