What's the state of the art in designing and creating your own life forms?
The problem is of course very complex so take my answers as simplifications.

Most transgenesis so far has been done on unicellular beings (bacteria, yeast), which we can change as much as we want fairly "easily", plants, insects and some mammals. Notably, for the latter case, mice are the species which has been used the most for transgenesis, as they are cheap (compared to, say, using pigs), do not take lots of space, have big litters and a quick reproductive cycle (~20 days). Aside from mice we've seen GFP rats, pigs, dogs, cats, sheep and probably there is more, but I will mostly restrict my answers to mice (as far as I know there are big technical difficulties in applying transgenics procedures to something bigger than a mouse, but to be honest I could not explain you the details on this as I have never done it).


  What features can be easily inserted into almost any organism easily?


Well, essentially, it is easy to give rise to monogenic traits (i.e. traits that depend on a single gene). Generally speaking, when you want to have more traits, or need more than 1 gene to have the trait you create single transgenics and then breed them together.

So, if your phenotype depends on the presence of gene A and B you will generate a (mouse) line expressing A, one expressing B, you breed them together and, if you are lucky, a part of the pups will be expressing both.


  Next, I imagine that to design and create an artificial life-form will consist of three steps: 1. Figure out which proteins will give rise to those feaures, 2. Design a genome that will express those proteins in the right quantities and 3. Implant that genome into a cell and actually let the being grow.
  Is the above sequence of steps correct, or am I missing something?


No, that is not how it is done!

There has been, to my knowledge, only one (very impressive) attempt to do that, a project led by Craig Venter at Celera Genomics. The project resulted in the creation of the first synthetic life form, named Mycoplasma laboratorium (see also the original paper on Science). Doing this for a multicellular being is, presently, science-fiction.

What normally you would do is to insert or remove just the piece of DNA you need in the genome.

There are different approaches to do that: for instance, you can start from embryonic stem cells, from fertilized oocytes or from spermatozoa.

This is the general process for making a transgenic mouse:


Source: https://web.archive.org/web/20161212170103/http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/T/TransgenicAnimals.html

Now, depending on how you inject your gene you can distinguish between transgenics and knock-in mice. In a transgenic mouse you insert the gene somewhere in the genome and, although there are method to understand where it ended up you generally do not know.
Knock-in animals, instead, target a specific region of the genome. This is usually used in order to change an existing gene, to improve its function or to remove its function (knock-out animals). This process, called gene targeting relies on the biological process called homologous recombination.

So, why don't we just synthesize a new genome each time?
Aside from Venter's dream, the process would:


have incredibly high cost
be extremely complicated technically
be extremely long (mostly for reason 2)
suffer from the problem that there is so much we do not know about genome regulation in a bacteria, let alone in a mouse! You have to think that the genome is not just a series of genes one after another: there is a big part of regulatory regions that define things like the 3D form of the genome, which determines when and how much genes are accessible for transcription, when and where they are activated and so on. Also, more and more attention is now drawn to something called epigenetics marks, modifications of the DNA, or its associated proteins that can also modulate its transcription.



  What is the cheapest experiment you can perform at your own home without access to a biology lab which involves changing the genome of a living organism in a way that's functionally visible? (An arbitrary example: Can you create an apple that's blue in color with a budget of $500? By create, I mean you should be able to hold the blue apple in your hand.)


Quick answer: no you cannot.


Health authorities will be not very happy if you start to produce transgenic organisms in your garage
The sole cost of the reagents which you will need to create the transgene will be >$500. You would then need various machines (e.g. a thermocycler, probably a spectrophotometer, a hood, etc etc.).
I personally don't have a clue how one creates a transgenic plant, but I suspect it is not much easier than an animal. And definitely you don't want a mouse colony in your garage, do you?


An easy experiment you can do is extracting (fairly unpurified) DNA with stuff you have at home. More than that it gets really complicated.