Will there be a problem running Time Machine in the back ground while using the Time Capsule's internal hard drive manually?
No.
Thus far, instead of using Time Machine in an automated mode, I've turned it on when I want to make a new back-up. Will it be necessary to run Time Machine in manual mode only or can I use its automated feature?
Or–I may be seeing the light–will running the two in tandem allow me to save each new back-up on both devices simultaneously, keeping both drives up to date?
I would suggest that you consider changing your backup strategy. Time Machine was designed to be an automatic "run and forget" operation that performs incremental backups of your Mac's internal hard drive. It will also backup an external hard drive attached directly to your Mac. Running it manually is not very effecient and you stand the chance of not getting a very recent (they run once an hour) backup when you need it.
For Time Machine I would suggest that you continue to use your external hard drive as is and dedicate your new Time Capsule for Time Machine backups. If you still need more external storage, you can always add additional external USB hard drives to the USB port on the Time Capsule.
Even better you will also want to include a second "full-clone" method to your backup strategy. This method will make a full copy of your Mac's internal disk drive that is bootable. Why do this? In case your Mac has a fatal hard disk crash, you can use the clone backup to boot your Mac back up and continue to use it within seconds. Without the clone backup, you would first need to install a new drive, then load a fresh copy of OS X, and then, perform a restore from Time Machine. This can take a while.
To perform these clone backups, I would recommend that you get another external USB hard drive with a capacity at least twice that of your current internal drive. I use Western Digital My Passport drives for this purpose, but any similar drive should do. To create the clone backup I would suggest using either SuperDuper! or Carbon Copy Cloner.
I would suggest that you create a new clone backup at least once a month or more often if your data is critical to you. Just connect a dedicated clone drive to any available USB port on your Mac and run the clone software to make the clone backup.