I just created a new development store to test some features after customer logged in
However I can't login/register as customer on the storefront
Every time I go to /account/login /account or /account/register it just 302 redirects me back to the home page
Does anybody know how can I register as customer in development store?
There's no restrictions that stop you from using a development store to login/register as a customer on the storefront. It's important to keep in mind that customer accounts are disabled by default, and I suspect this is why you're getting redirected to the homepage when trying to use the login or register pages.
You can enable customer accounts in the Shopify admin under Settings > Checkout > Customer accounts. Choose either 'Accounts are optional' or 'Accounts are required' and then make sure to save your changes.
Hopefully this helps!