So,
  I played this for the first time today.
  Is it normal to roll d6 for Love, Fear, Self-Loathing and Weeriness?
  And, if so, is it normal to discard 5s and 6s?

  If not, and I am starting to suspect it is not, what do you think the impact of this would be on the game?

Hello Dave

What do you mean by "normal"? According to the game text, you're supposed to roll d4's, discarding the result "4".
You're certainly changing some probabilities around if you're playing the "d6, discard 5 and 6" way. What dice do Intimacy, Desperation and Sincerity get in your version?
This thread and this one dating from the playtesting may provide some information as to why the rules are the way they are today.

The base mechanic is essentially rolling d4's and discarding 4's. This means each die produces a result of 0-3. Each die is equivalent to a d4-1.

Really, it'd just a standard Nd4 curve, but throwing away 4's makes the arithmetic a lot easier. Which is easier to add:  1 4 3 2 1 or 0 3 2 1 0?

The 0-3 range for each die also makes the Intimacy d4 (1-4), Desperation d6 (1-6), and Sincerity d8 (1-8) worth that much more.

Playing with d6es and discarding 5's and 6's is a slightly different result curve, as the results on each die are {0,0,1,2,3,4} (mean 1.67) instead of {0,1,2,3} (mean 1.5). However, the range is quite different on the top end, since 4d{0,0,1,2,3,4} ranges from 0-16 but 4d{0,1,2,3} ranges from 0-12. What this accomplishes mechanically is to reduce the power of the Intimacy/Desperation/Sincerity mechanic by a couple points.

Will this ruin play? I don't think so. All the rolls in the game are opposed rolls anyway, so you're rolling weird new odds vs. weird new odds. If you're concerned about changing the I/D/S probabilities, shift each one of them up a die size (to d6/d8/d10).

Sometimes I have trouble finding enough d4's. Some people hate rolling d4's. So I can see the appeal of using d6 instead.