Crankset arms length and chainring for 5 year-old My five years old little girl rides a BMX micro mini bike with 18 x 1" wheels. The original bike came with 130 mm crankset length, 38 tooth chainring and 14 tooth rear cog.

However, looking at her on her trainings, came to my attention the way she was pedaling. After reading a lot in the web I found out that there are shorter  crankset for small rider. Due to the fact that I can´f afford $150 US dollars crankset plus shipping to my country, $70 US plus customs fee. I decided to cut and shorter a 175 mm crankset arms to 110 mm, and I did a great job. 

So, due to your expertise and knowledge, I would like to ask you which chainring and cog could you recommend for my daughter.
Gearing is a very subjective question.

Using Sheldon Brown's Gain Ratio Calculations, 36:17 on 110mm cranks will give a 1.25% lower gain ratio than 38:35 on 130mm cranks.