Question
1. is the bottom half of the stem supposed to get brown early on in the summer or at all?

2. when the flowers die, will more bud if you cut off the dead ones?

Answer
Hi Steve,
Thanx for your question.  The bottom half of the stem of your tiger lilies should not be turning brown this soon.  This should happen towards the end of summer as the foliage dies off.  If the plant continues to turn brown, dig up the bulb to check for rot.  If the bulb is rotting or smelly, throw it away (in the rubbish) and do not plant anything in the same spot for a couple years.  

Once the lily has bloomed, cutting off the spent bloom will not produce more blooms.  The lily blooms once in the season.

I hope this helps.
Tom