You may have to wait 30-45 days to app for a Best Buy card since it is Cap1. I think you will only get $200-$500 if approved.
Unfortunately, I don't see a high credit limit approval in your near future. It will take time for that. The only way to do that now would be to get a secured card and load it up with $5,000 of your own money. But as you build a good long payment history, you will start to get CLIs, and then when your score is on par with prime lenders like Discover, AMEX, or Chase, they will approve you for a card that matches or slightly exceeds your average credit limit and then you establish history with them and get rewarded with a CLI and then you repeat the process until you have the limits you desire. This is an embarkment on a 2-3 mission and during your journey, you may have to close out accounts that aren't growing with you.
For a mortgage, most lenders want to see 3-4 active trade lines and most lenders require a 640 mid fico score. There are a few lenders that will approve a 620 but they are becoming much harder to find and there are more hoops to jump through for approval. Below 620 fico approvals are becoming a thing of the past and starting June 3rd, all below 620 mortgage loan applicants will require manual underwriting and a recommendation from their LO/Broker. Also after June 3rd, PMI will be required for the life of the loan and will no longer be cancelled at 78% LTV. You will need to have an FHA case number by the end of May. Getting that requires that you are first on contract for a home. I encourage you to look into NACA. Their website is
www.naca.com They will approve you without a basis on credit score. They approve based on your ability to repay, which is determined through satisfying 14 pages of requirements and attendance of their seminars/classes. NACA loan approval takes between 3-18 months depending on how quickly you can satisfy all of their requirements. There are a few treads about this program on the Mortgage Loans forum.