Yes, you can use the same Apple ID on the iPad that you are using on the iPod. That is the ID that you created personally so it makes sense to use that one. You do not want to have two ID's - and in this case I mean two personal ID's - if you want to share content. When you share an ID on iDevices - that is what allows you to download apps on each device without having to pay for them again. If you live in the US, you can download virtually all content again for free except for some movies, audio books and Extra features content with movies in iTunes.
So that you can control what downloads to your child's iPod - make sure that you do not turn on automatic downloads. Settings>Store>Automatic Downloads>Off for apps, books and music. If you want the child to have something that you downloaded, you can always download it again on the iPod or if you sync with iTunes - just sync it to the iPod when needed to transfer the content.
Yes you can absolutely choose what you want to sync to all iDevices and iTunes will remember the sync configuration the next time. I do not use auto sync at all. I always initiate the sync on my iDevices when I am ready to sync. On a PC - launch iTunes and go to Edit>Preferences>Devices and there is a box at the bottom - prevent iPods, iPhones, iPads from automatically syncing. Check that box, click on OK and quit iTunes.
Connect your iDevice to your computer, launch iTunes and then click on the iDevice name on the left side under the devices heading. Then you can go into each tab in the iTunes window on the right and select the exact content that you want to put on each device. After you make your selections, click on Sync or Apply in the lower right corner of the iTunes window. That will lock in that sync configuration on the device and iTunes remembers that each time that you sync.
If you want to continue to auto sync, go back and turn it back on by using the steps that I described above.
Here are three articles that talk about syncing with iTunes that will be helpful for you to read. The user guide is also included as a bookmark in the Safari browser and I included a link to the PDF version of the manual.
http://support.apple.com/kb/HT1296
http://support.apple.com/kb/HT1386
http://support.apple.com/kb/HT1351
http://manuals.info.apple.com/en/ipad_user_guide.pdf