You got the first part right 🙂
For the second part:
Remember that if the market rate was the stated rate, then the bond would sell at par (face value).
Since the bond's stated rate is above the market rate, we would sell the bond above par (face value)
How does that work?
Well technically, if the stated rate equaled the market rate, then the present value of the face amount + the present value of the total interests payments would equal the face amount (in present value terms).
Since the stated rate is greater than the market rate, the interest payments will be greater than they would be if the stated rate equaled the market rate. Therefore, when we discount the total interest payments using the market rate and add that to the present value of the face amount, the sum will be GREATER than the face amount.
Let's use your original example and restate it with the market rate equating the stated rate
A company is issuing 10%, 5 year, $1,000 bonds. The market rate is 10%.
Present Value of $1000 at 10% for 5 years is $620.92 ($1000 x 0.62092)
I used the table for present value of 1 and looked at the coefficient for 10% at 5 years (.62092)
Present Value of the total interest payments are $379.08 ($100 x 3.7908)
Interest payments at $100 per year (10% x $1000)
I used the table for present value of an ordinary annuity of 1 and looked at the coefficient for 10% at 5 years (3.7908)
Therefore the present value of the future cash flows are $1000 ($620.92 + $379.08), which is the face value of the bond
Now, let's use your original example with a stated rate that is greater than the market rate:
A company is issuing 10%, 5 year, $1,000 bonds. The market rate is 7%.
Present Value of $1000 at 7% for 5 years is $712.99 ($1000 x 0.71299)
   I used the table for present value of 1 and looked at the coefficient for 7% at 5 years (.71299)
Present Value of the total interest payments are $410.02 ($100 x 4.10020)
   Interest payments at $100 per year (10% x $1000)
   I used the table for present value of an ordinary annuity of 1 and looked at the coefficient for 7% at 5 years (4.10020)
The present value of the future cash flows is $1123.01, which is what the bond should sell for. As you can see, the bond's present value is greater than the face value of the bond; so it should sell for a premium.
Hope that helped!