Re: Is it common in american english to drop r's when reading?
It really depends where in in the country your are.
In New England, the r's after vowels often were very faint or dropped all together. My name is Barb but after living there for 15 years, I answer to "Bob."
In other areas, you will find the final r sound in words that end in "er" sound like "uh" - mother is like "muthuh."
That's why I often laugh when people say they want "an American accent." There are dozens. (Same with a British accent.)
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.