No, they don't. The EU parliament is not a legislative body.
Wrong. The Parliament is the legislative body, together with the Council. That's no big difference to countries that have a bicameral parliament. The main difference is that in the EU, the term "parliament" only refers to the lower house whereas in bicameral systems, the term refers to the system of both houses.
What is missing (for both Parliament and Council) is the right of initiative, which is reserved for the Commission. That is an issue in theory. In practice, however, it does not make a big difference as even in countries where the parliament has the right of initiative, the proposal for new laws come from the executive branch anyway.
Policy may be suggested by the commission. The parliament gets to say yay or nay to that. They also have a say in who gets to be appointed to the commission.
The policies decided by the council are then made directives by the commission, to be implemented by the member states.
That's completely wrong. "Directives" are legal acts proposed by the Commission and voted upon by the Parliament and the Council. The title even says "Directive (EU) year/no of the Parliament and the Council". (Same for Regulations.)
The parliament is basically a fig leaf, the only part of the EU bureaucracy that directly represents the voters in the EU. But its competences are very limited. As a voter, you have more influence by voting for your country's parliament.
That's also wrong. If you look at the legislative competencies of the European Union, that's no longer true. Many basic decisions are now made by the European Union, and thus significantly influenced by the European Parliament.