What are evolutionary implications of contraception and reduced childhood mortality rates worldwide?
Before the introduction of contraception, humans conceived quite a lot of babies (there was little to do to avoid that), but the population was kept in check by very high childhood mortality rates. This mortality rate ensured that only the fittest survived.


Woh! Let me stop you there. No, high rates of childhood mortality do not ensure that whom Natural Selection is acting upon are the "most fit." That title is given to individuals who are able to survive long enough to breed. 

Many, many conditions that affect humans do not interfere with a person's growth or ability to perform physical or intellectual feats - but can definitely reduce or completely eradicate a person's ability to reproduce. These individuals are evolutionary dead-ends, and thus - while they survive in the environment - are not "fit" in the context of this topic.


  What are the implications of reducing childhood mortality rates worldwide on human evolution? Maybe evolution is a wrong term to use here. Would it be fitness as species?   


What it means is that populations will rise. As populations rise and intermingle, alleles get spread around. Perhaps rarer alleles become more common, or more common alleles become rarer. We're talking about Hardy-Weinberg equilibriums in the allele frequencies. However, with populations large enough true equilibriums are impractical to reach.

The more pragmatic explanation is that more mutations will arise, they'll be passed around, rare mutations will see more exposure, and humanity - in a very general sense - becomes more diverse genetically until a Selective Pressure is applied. Until there's a detriment to having a particular gene, there isn't a need to eliminate it from the population.

Your example is children who are too weak at birth to live outside of an NICU for their first few months - well, because we're capable of providing such care, until that care goes away it's not negatively affecting us. If it comes with an additional cost - say that, for whatever reason, premies can't digest mother's milk until the 8-month mark, then Natural Selection would pressure humans to evolve so that only 8+ month or older babies are born.

There's no evidence of significant pressures in human populations at this time, though. At least not in politically stable and technologically apt countries, although in areas undergoing long-time drought or other dangers it is seen. Sickle-Cell Anemia is though to be an adaptation to rising Malaria rates - which was quite fatal until medication was devised to treat it in the last 50 years or so (not even a blip in evolutionary terms).  


  Is there any evidence that people who lived in a world without contraception and with high childhood mortality rate were more physically or mentally fit?


Not that I can think of or find easily. While there are detriments associated with being born prematurely, and some evidence that women waiting until they're late 30's and men until they're 60 or older can negatively affect their gametes and consequently the development of any children born from them - it's been such a short time since the introduction of hormonal contraception and modern medicine that we may not see results tangible results for hundreds of years. Although condoms have been around for several hundred years, and there's been no associations with condom use that I can think of.

In the short term, what it has done is affect the ethnic diversity of countries. For the first time ever, Hispanic births have outnumbered Caucasian births in 2012 in the United States. Other countries, mostly European countries, are seeing declines in birth rates - which means their populations will decline with age or be compensated for by immigration. Higher birth rates, however, are associated with lower income brackets or very religious communities - both of which are associated with ethnic Minorities, at least in the United States. So while minority populations have more kids, they do so on fewer resources which may negatively affect their children in the long-term, whereas higher-earning segments of the population might have fewer children later in their lives, but can provide a much more stable environment and opportunities to continue that success.

One thing that hormone-centered birth control has done, however, is eliminate rape as a viable form of passing on one's genes. As gruesome as it might be to consider, pregnancies as a result of rape can produce "fit" offspring. It is a legitimate reproductive strategy in nature, and is in humans... unless the woman is using contraception. In the long-term this will probably show some interesting results (nominally an enhanced "Female Choice" effect - which is already evident in human evolution), but nothing right now.

It's also worth considering the ability for humans' increasing abilities to manipulate our own genome. At this point, we're nearly able to remove detrimental genes at-will and replace them with more effective versions. Because of this bio-engineering ability, the generations after ours might never be susceptible to inherited diseases - or they might all hop on the trendy "genes" and end up completely identical to each other. That's part of the fun of being human, though; we're very messy when it comes to Evolution. We tend to muck things but up good. :)