"Is there a way to output the billing_region & billing_country select fields from Store"
Yes and No.
Yes you can:
{exp:store:checkout disable_form="yes"cart}
<select name="billing_country">
{billing_country_options}
</select>
{/exp:store:checkoutcart}
EDIT: I originally incorrectly suggested {exp:store:checkout disable_form="yes"} - this is an invalid parameter.
Or just the raw values:
{exp:store:cart}
{billing_country} = {billing_country_name}
{/exp:store:cart}
Which can work inside your {exp:user:edit} tags.
But the answer is also No - it will only work when the user has an item in their cart, so if they come back at a later date, or have registered prior to adding products into their cart, the tag pairs will fire the {if no_results} conditional and nothing will get output :(
Other routes - you could use a third_party country/region listing plugin such as Reegion Select by Derek Hogue. But then you may have differences in the references for the countries & regions, which may be important for your payment gateway and syncing with Store on Checkout.
Or I've written a plugin that taps into the Expresso Store classes and outputs the regions or countries independently of the {exp:store:checkout} tags - On Devot-ee.
"the checkbox for setting the shipping info to "Same as Billing Details" is checked and overwrites"
You'll need to force this in your template, something like:
{if "{shipping_country}" != "{billing_country}"}
<input type="checkbox" name="shipping_same_as_billing" value="0" id="deliveryCheckbox">
{if:else}
<input type="checkbox" name="shipping_same_as_billing" value="1" id="deliveryCheckbox" {shipping_same_as_billing_checked} >
{/if}
If you find that the shipping details are still being incorrect displayed in the checkout form data being generated by Store, then you could output the shipping details from the member details instead:
{exp:member:custom_profile_data}{shipping_country}{/exp:member:custom_profile_data}
Update
...{exp:member:custom_profile_data} tags and get the same results either way. My custom member fields are named the same as the Store fields...
You mention your member fields are named the same as the Store fields, that means that the store tags will be parsing the fields first before the member tags get a look in. So you want to change the parse order to get the member tag to kick in first, so use the parse="inward" parameter.
{exp:store:checkout form_class="checkout_form" form_id="checkout_form" next="store/order-confirmation"}
...
{exp:member:custom_profile_data parse="inward"}
...
<input type="text" name="shipping_address1" value="{shipping_address1}" data-msg-required="Please enter your shipping address." required>
...
{/exp:member:custom_profile_data}
...
{/exp:store:checkout}