Should I use an int column to represent date?
Shorter answer to the title of the question: No. Why would you want to lose the ability to treat a date like a date. So important for sorting, date functions, etc.

A few thoughts to get you started at least, not sure which DBMS while answering, answering based on my experience with SQL Server:

1.) GUID as a Primary Key is generally not a great idea. Not in SQL Server especially. What's wrong with an Integer primary key? Yes whatever your Primary Key is becomes your Foreign Key in the child table(s) so that is large and probably not necessary.

2.) I would use a regular datetime column. Performance wise, if you are well indexed, you shouldn't notice a difference here. And a datetime column is more versatile as, well, a date. You can ask questions of it that you can't ask as easily of an INT column with built in datetime functions. If you don't need time and you are on a DBMS or version that has just the DATE you can use that data type.

3.) Yes. Especially if CustomerID is a foreign key to a Customers table. Good to index. Whether or not you need to make an index on CustomerID AND the Date column depends on how the queries will typically look. If you are often querying joining to customer and specifying a date range you may find it beneficial to have the date. You may find it beneficial to include some other columns to cover other queries as parts of the key or included columns. It really depends on your queries and data, though.

As far as clustering on the date column. That's a hard one. If this were a fact table in a warehouse and every single query was always on a date range, then there are some benefits there. If this is an operational invoice table, I imagine your app also joining into invoices in other approaches also. I also imagine invoices being queried by invoice IDs stored in other tables, etc. So I don't believe there is enough to determine clustered key. I'm of the school that prefers a simple surrogate key for OLTP tables. An InvoiceID INT (or BIGINT if you really would blow out an INT) setup as an identity column so it is always increasing and avoiding page splits. But I don't know if there is a definite wrong answer here (well there are many, but you haven't proposed any of those)