We can see more distance than the age of the universe Because Expansion of the universe
Here is a complete explanation
https://medium.com/starts-with-a-bang/how-is-the-universe-bi...
For example, if we heat up a metallic sphere, then it will expand in all directions. If we project this expansion backward, then we will see that metallic sphere was created just few hours ago. Do you agree with that?
Moreover, why I should accept that Universe is expanded in single whoooping event? IMHO, constant vibration (noise) of gravitation field explains same effect without need for this single event. Vacuum just contracts and expands by tiny bit all the time, because something happens in Universe all the time, so light ages as it travels.
To take your analogy, we witnessed the expansion of the metallic sphere from a point too small for us to see, there may be more going on that we don't understand, but it certainly looks like it expanded out of nothing.
(1) - wrong, Hubble "constant" has different values depending on frequency of light or radiowaves used for measurement.
(2) - wrong, but papers about this started from 2008 only. In short, increasing diameter of telescope makes galaxies look "older".
Stephen Hawking's The Universe in a Nutshell does a nice job of introducing many of these concepts for those interested.