For the mattress, get more life out of it by putting boards between the mattress and box springs.
For the computer, when mine died, I ordered a refurbished laptop from Tiger Direct for less than $100. Been using it two years without problems. Perhaps that's something you might consider for the short term.
To amass dollars for a down payment, get in the habit of asking for bonuses. I just did 8 gas stations at an average of $57.50 each that started on the job boards at $10. But like you stated earlier, where competition is fierce, shops tend to disappear right off the board at base pay. And many of the more lucrative shops just aren't being done currently.
I live in a super rural area, and there literally is not another shopper within 100 miles. Unfortunately, there aren't many shops much closer than that. I HAVE to ask for bonuses to make driving 200 miles worthwhile; and then, to be profitable, I have to turn the shop into a mini-route.
When I started doing this, I was a lot younger, and had the energy to shop all the thrift stores when I was on a shopping route. I would buy stuff at the thrifts, and sell on e-Bay and Amazon. I picked up magazines, books, china, Aloha shirts (OMG, Aloha shirts!! Buy for $2 or less, sell within 5 hours for $25-$75!!!), stuff like that. I bought a gravy bowl for fifty cents, and sold it for $105.50 on e-bay. I got a Sears catalog, 1919, NOT facsimile, the original, for $2 and sold it for $100. I picked up some sci-fi magazines for free, and one of them had the original short story "The Fireman" (Fahrenheit 451). I sold that issue for $100, and sold the rest of them on Craigslist.
Just an idea; that's my method of "double dipping".
You might also consider doing shops that don't pay much, but reimburse in full for stuff you'd pay out of pocket for. Groceries, oil changes, stuff like that. When I was in Salt Lake City, I did a LOT of grocery store shops for Trendsource. Didn't pay much, but I never had to spend money out of my pocket for groceries. That saved me a fortune. And since most of the pay was reimbursement, no taxable income created.
Best of luck.