If the 1099 MISC is in the name of the business, it gets entered on form 1120 S only.
Quarterly estimated tax payments are entered on form 1040 only.
The S-Corp does not pay income taxes. The earnings of the S-Corp are distributed to the shareholder. So, the income is recorded on the S-Corp return. It is distributed to the shareholders after expenses are deducted on form K-1. The shareholders must pay tax on the distribution. Estimated tax payments are made by the shareholders to pay for the taxes due on S-Corp distributions.
