Is it possible to left align the rows per pages and page navigation and put a button in its place in the same row?

The actions row are aligned right by default. There is a way to achieve what you want though. You can use custom pagination and hide the current one:
<v-data-table
:headers="headers"
:items="desserts"
:search="search"
hide-actions
:pagination.sync="pagination"
>
And add this after the datatable:
<v-layout row justify-center>
<v-pagination v-model="pagination.page" :length="pages"></v-pagination>
<v-btn class="test">test</v-btn>
</v-layout>
See more here.
Here is a Codepen in action.
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