Question: I find when playing in a minor key I always end up playing the major version of the V chord. (So in Em I always end up playing a "B major" chord.)

However, the third of this chord should be played as a minor when looking at the key and it seems like using an accidental as frequently as I do is not really in the spirit of accidentals.

Am I making a mistake in my music theory? Am I playing the key I think I am? Is this just a mode? Do I need to start playing more B minors?

Please bear in mind I am relatively new to music theory and would appreciate it if you treat me like an idiot. Thanks.

Answer: 
Although the D in the scale of B will be natural in the key signature, it will often be sharpened when leading to the key note E. Think of the melodic minor scale for E. D will be sharpened on the way up but natural on the way down. 

So, if your chord is to resolve to the tonic E, the sharpened D will work nicely. You might find that a B minor chord works well if it goes somewhere else next.
Yes, you are making a mistake in the theory you know. Each major key sig. has a relative minor to go with it - C- Am: G- Em; Bb- Gm etc. Whilst the majors adhere exactly to the # or b in their key sig., minors cannot and do not.

That's because the notes that make up the minor scales vary in their 6th and 7th notes. It's the way they are. Take A minor. 6th note can be F or F#, 7th can be G or G#. In your scenario, E minor can have C or C#, and D or D#.

The D/D# anomaly makes the chord based on B either B minor or B major respectively. Both are quite acceptable, and both are used. The B major, as you find by using it and it sounds good to you, uses the leading note of D# to sound more decisive going into E minor, but there's nothing in the theory that says we must use that, or we must use B minor. Whatever fits better is what we should use. Please don't use theory as a set of *rules*. It's more a set of *guidelines*. Usual adage - if it sounds good, it usually is - whether the 'rules' get broken or not!