Is there a bottom limit to an optimal body fat range that it would be considered unhealthy to be below? In other words, can you have too little body fat in modern (1st world) living conditions when it is unlikely to need to have stored body fat for unpredictable economic conditions (famine, war etc)?
Yes, absolutely. Body fat is essential, and not just a form of stored energy. Symptoms of excessively low body fat include hormonal disruption (low testosterone in men, amenorrhea in women), loss of bone mineral density , loss of muscle, weakening of the immune system , and cognitive impairment. 
Quoting Dr Eric Helms, in 'The Muscle and Strength Pyramid: Nutrition':
Make no mistake, there are substantial and important differences between being a recreational bodybuilder or lifter, and a physique competitor or strength athlete. A competitive physique athlete has to achieve a level of leanness in order to get on stage that often results in the loss of a normal menstrual cycle among women, a decline in testosterone in men, disrupted sleep, higher levels of stress hormones, hunger hormones, a decline in satiety hormones, a reduction in total daily energy expenditure disproportionate to weight loss, decreases in lean body mass and strength, higher levels of fatigue, and is often associated with a diet that carries a higher risk of micronutrient deficiencies [1-9].
- Helms, E.R., A.A. Aragon, and P.J. Fitschen, Evidence-based recommendations for natural bodybuilding contest preparation: nutrition and supplementation. J Int Soc Sports Nutr, 2014. 11: p. 20.
- Halliday, T.M., J.P. Loenneke, and B.M. Davy, Dietary Intake, Body Composition, and Menstrual Cycle Changes during Competition Preparation and Recovery in a Drug-Free Figure Competitor: A Case Study. Nutrients, 2016. 8(11).
- Fagerberg, P., Negative consequences of low energy availability in natural male bodybuilding: a review. Int J Sport Nutr Exerc Metab, 2018. 28(4): p. 385–402.
- Hulmi, J.J., et al., The effects of intensive weight reduction on body composition and serum hormones in female fitness competitors. Frontiers in Physiology, 2017. 10(7): p. 689.
- Rohrig, B.J., et al., Psychophysiological Tracking of a Female Physique Competitor through Competition Preparation. Int J Exerc Sci, 2017. 10(2): p. 301–311.
- Petrizzo, J., et al., Case Study: The Effect of 32 Weeks of Figure-Contest Preparation on a Self-Proclaimed Drug-free Female’s Lean Body and Bone Mass. Int J Sport Nutr Exerc Metab, 2017. 27(6): p. 543–9.
- Rossow, L.M., et al., Natural bodybuilding competition preparation and recovery: a 12-month case study. Int J Sports Physiol Perform, 2013. 8(5): p. 582–92.
- van der Ploeg, G.E., et al., Body composition changes in female bodybuilders during preparation for competition. Eur J Clin Nutr, 2001. 55(4): p. 268–77.
- Maestu, J., et al., Anabolic and catabolic hormones and energy balance of the male bodybuilders during the preparation for the competition. J Strength Cond Res, 2010. 24(4): p. 1074–81.
Yes. Bodybuilders frequently aim for body fat percentages in the single digits in order to look "ripped". While it is acceptable for short term (such as a competition), body fat under 5% (for men, and about 8-9% for women) can lead to long-term health problems (below 3% is fatal for humans). One such problem is bradycardia (too slow of a heart beat) leads to dizziness, syncope, passing out and cardiac arrest. Vitamins A, D, E & K are fat soluble vitamins. The sort of diet necessary to reach sub-10% body fat will be extremely low in fat and by necessity will be deficient in those vitamins. Since D is necessary for calcium absorption, brittle bones are an effect of very low body fat. Other negative effects include "brain fog" due to insufficient essential fatty acids that our brains require for proper functioning.
Percent body fat declined from 14.8% to 4.5% during preparation and returned to 14.6% during recovery. Strength decreased during preparation and did not fully recover during 6 months of recovery. Testosterone declined from 9.22 to 2.27 ng/mL during preparation and returned back to the baseline level, 9.91 ng/mL, after competition. Total mood disturbance increased from 6 to 43 units during preparation and recovered to 4 units 6 mo[nths] after competition.
Studies of longer term effects of very low body fat have shown depression, loss of strength and lower testosterone levels that can take months to recover. The "super ripped" appearance of body builders is no longer appealing to me (it was in my 20s).
My advice (for cis-gender men) would be to stay between 10 & 15 percent body fat. Very few people are willing and able to maintain the effort necessary to get lower. The human brain is about 60% fat. I'm a software developer and my brain is my most important body part.
Body fat pads and insulates vital organs:
Among its many functions, fat surrounds and cushions vital organs like the kidneys and insulates us against the cold.
It also serves as the longer-term store for needed calories. The same article talks about how the brain uses an extremely high percentage of the body's daily caloric intake (up to 1/3). It's not difficult to understand why having too low body fat would impair thinking.
Can you have too little body fat?
"According to the American Journal of Clinical Nutrition, there are healthy body fat percentages based on your age. For people aged 20 to 39, women should aim for 21% to 32% of body fat. Men should have 8% to 19%. For people 40 to 59, women should fall between 23% to 33% and men should fall around 11% to 21%. If you’re aged 60 to 79, women should have 24% to 35% body fat and men should have 13% to 24%."
Yes. Lack of fat can kill you. This is sometimes known as "rabbit starvation" (also known as "Protein Poisoning") because in some cases people whose only source of food was rabbit meat (which is very lean and lacks sufficient fat for a human to support themselves on) became sick and ultimately died.