Can two visas be applied at the same time?

If you already have a valid visa for the middle east, and you are in possession of your passport - there is nothing stopping you from applying for a visa to Japan.

As far as I am aware, Japan does not have any specific restrictions from visitors holding middle eastern visas (or against middle eastern nationals).

To apply for most visas you need to send in your passport. In some exceptional cases you can request your passport be returned while your application is being considered but this is rare.

As a general rule you can have multiple valid visas in your passport. I have five valid visas currently in my passport.

There are a few exceptions, where country A will not stamp a passport with country B's visa or entry stamp. But these tend to be neighboring countries in conflict with each other.