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Both Disc and BOA do allow additional deposits at any time. However, BOA will HP to add secured deposits, so best to do as much as possible at one shot. I added to my discover a month before its graduation (from 600-2k) they did look at credit (as the CSR said) but it was a SP to add funds.
1) I had BofA secured, you can CLI at anytime, each time it will be HP.
If I have to start again, I'll get the card with $1,000 or more, use it, PIF before due date, after 5th statement extra deposit to make it $5,000 or more (you can go as high as $10,000). There is a big chance for auto-graduation at month 9. If you don't get graduation after 9th statement call and ask for one, latest reports say now is SP. Apply for other cards after month 9 (3-4 months report with 5k limit)
Check if you pre-qualify for Amex (check the site after your card is reporting 5k limit). Wait for a good offer (low APR, extra bonus) and get a card with no AF. The limit will grow fast.
Don't get store cards, or only 1 or 2 that you really use. DW JCP card grew from $400 to $6,500 in less than a year, is the only store card she has
2) BofA won't do this, you keep the same card
3) I haven't read a single post saying they have $1000 and graduated with $500. All stayed the same or got CLI at graduation.
Good luck