Possible to allocate input to specific cells or more than past the first row?

I’m afraid I don’t fully understand your question – so I’d appreciate if you could explain it in more details.
A few basic facts about current implementation:

• Field names should be in the 1st line.
• New submissions are appended to the next empty line (2nd, 3rd and so forth).

Sorry it may sound a bit messy.

What I mean is, is there any way to allocate a form to a specific cell, such as if the form field name is ‘Address 1’, then it would go in E6 where I have the ‘Address 1’ column in my database.

Sorry – no.
But you can populate all form’s data in a separate tab. And then “reference” specific parts of it from other tabs – as described in https://support.google.com/docs/answer/75943?hl=en

I didn’t think of that! That may solve my problem. This is perfect, thanks. Do you have a pro version or something for this plugin?

No pro version for a meanwhile – just a regular one.