[SalesForce] Can a community user create Account in community page

No this is not possible. When you create a customer community user, the user's role is determined by his/her contact's account owner's role.

This determines the level of access the customer community user has on the contacts of his account and other related records. If you change the account on the contact, then the whole roles, access and sharing or related records need to be revoked from the old account and need to be created in new account which is cumbersome.

I could find this documentation mentioning about the roles.

The available profiles for external users are limited to the Partner Community User profile or profiles cloned from it. Unlike a standard user, the partner user role is automatically assigned based on the account name. The partner user role is a subordinate of the account owner’s role. Therefore, all data for the partner user role rolls up to the partner account owner’s role. If you disable a partner user, the partner user role becomes obsolete, and the data no longer rolls up to the partner account role.

Does the user have create and edit access on accounts & contacts?

I would try using a create contact custom action with the Account filled in with a merge field for the current User's account. This will save the user the trouble of having to fill in the field, and you can guarantee it is the right account on the Contact.

From there, I would try to use the predefined values on the custom action to set the Owner field as well. Since you are associating the new Contact with an existing Account, you will just need to add the current User's Contact Owner or the current User's Account Owner, (or whatever value you do so choose) as the New Contact Record's Owner.

Let me know if that works for you.