How is in-vitro fertilization technically achieved?
After following user David's hints, I have collected some videos that answer the questions I posted:


Regarding to needle manufacturing, there exist machines able to create micropipettes (this is how the needles are actually called) by heating glass cilinders and pulling them to create very thin tubes (capillaries). Click for featuring video.
Regarding to the high-precision manipulation, technicians use mechanical devices able to convert macroscopic motion (eg: human manipulation) to microscopic motion (eg: using reduction gears). Click for featuring video.
Regarding to how they keep the egg still, technicians use another type of pipette that slightly "sucks" the egg in a vaccuum-cleaner style. For a full-feature video of how the egg and sperm are manipulated, check this video out.