How is this map different from a typical us map?
what do all of the new names on the map have in common?
how do you think this map influenced established immigrants who saw it in the 1860's?

i have answered the first two parts of the question. i need with the last part. here is what i have:
the map above is different to other maps of the u. s. all of the states have crossed out their proper names and written in irish ones.

1, In this map all state's names are redacted and rewritten.

As you can see, the name of the state in the map are crossed and rewritten with a name that would not be familiar if mentioned to local citizens of united states. This indicates that other people beside citizens of united states were using this map.

2. The common thing is that All of the states now have Irish names.

Now we know that this map is created for Irish readers not local citizens. As you can see, name such as o'toole , Mulligan, and shaughnessy  that are used to replace the name of the state is a very common family name for people in Ireland.

3.  it helped the irish immigrants that saw it because it is easier for them to read and appealing to the immigrant eye

This map was created during the great famine era. At that period of time, people in ireland experienced massive hunger problem due to various factors such as war, inflation, and crop failure. This make many irish people decided to find better opportunities in united states.

i think they would live in caves or natural tunnels