What are some features that do not make an individual better than others without this feature?
Phenotypic effect of mutations

Most mutations have no phenotypic effect and therefore no fitness effect either (fitness is a function of both expected survival and expected number of offspring).

These mutations that have no phenotypic effects are therefore all neutral in terms of their fitness effect. Their probability to be passed on to at least one offpsring in a panmictic population of constant size $N$ is $\left(1-\frac{1}{2N}\right)^{2N} ≈ exp(-1) ≈ 0.36$. A mutation that is beneficial would have a greater probability and a deleterious mutations would have a lower probability. There are a whole bunch of equations for these but I don't want to get too "mathy" in this intro post.

Fitness effect of mutations

Among mutations that have phenotypic effects, some have no effect of fitness, some are deleterious, some are beneficial. The beneficial ones only represent a very small minorities of mutations.

Note that it is important to make the difference between a trait that seems to not be beneficial to you, while whether it actually is or is not beneficial. You cite things like eye color and head shape. Both of these traits can have very important on sexual selection. Try to date someone with a triangular head or with white iris! Also, if I am not mistaken, light colored eyes tend to let more light going through to the retina, hence light colored eyes are better adapted in environment where sunlight exposure is low. Head shape may have important impact too. For example, the jaw bones is an important contributor of the head shape and whether you have a strong or weak jaw is of importance (and the shape of the jaw is under sexual selection pressures).

Examples of neutral mutations

Most mutation that happen at non-coding, non-regulatory regions of the genome will be neutral.

Among the mutations that have phenotypic consequences but no fitness consequence, there are classical genetic markers (such as tongue rolling) that are likely free from any selection pressures. Note, that heritability for such trait is rarely 1.0 (see here for explanations). Maybe earlobes would be a good example, although there might have some sexual selection going one here.

Note also that as many mutations are recessive, the carrier of a single copy of the mutation may not be affected.

What is a gene?

You are using the term gene improperly. A gene is typically defined as a coding region. You might want to study a little the basis of genetics here. You should typically get used to the terms locus, allele and gene copy number. This answer can help you at this endeavour. 

A small comment on the frequency of mutations


  If a mutation occurs [..]


Just to make sure you do not have a misconception about how frequent mutations are here, I would like to highlight that, in humans, at every reproduction event, the new born has about 10 to 100 new mutations that were not present in any of both parent. So, mutations are rather common!

Introduction to evolutionary biology

Your question is a very introductory question to evolutionary biology. You might want to have a look at a short introductory course such as evo101 by UC Berkeley.