No Mutants Allowed

No. AI needs giant machines the size of rooms in Fallout.

I am in doubt about this as there were androids in 50s sci fi (as far as Metropolis). I think it is the way how it is approached, the Synths could have worked but the concept is wasted in Fallout 4

Not the way Bethesturd implemented them.

No, and the "androids" Dutch refers to is in cinema only, However clunky dumb robots work out just fine. The reason Synths don't work is because of the future tech creep in an environment where resource acquisition is basically or nearly impossible to obtain the materials. Also transistors.

The AI computers Zeno mentioned are pre-war. It does make sense for synths to exist in Fallout. But given how pre-war computers were build with kooky 1950s tubes and such, it would take humanity longer to develop compacted AI such as the synths. Probably the same timeframe as now (we're building AI!) from our 1950s. Maybe longer cause of post-apocalyptia obviously.

Short answer: No.

No.

Long answer: Nooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo.