Is a foreign child (age 5) automatically a U.S. citizen when adopted by an American?

No. First, the orphaned child must be legally adopted in its country of citizenship, and that adoption finalized. Due to the high incidence of child trafficking, baby sales, and fraudulent immigration of relatives through "adoption," various countries have enacted protections or banned foreign adoption completely. Many countries have a sort of "trial period" for the child to live with its adopting parents & then a determination by child welfare services of what's best for the child before it is final. Once the parents have the final court order of adoption, they take the child to their country's consulate to begin the immigration process for that child. 

It is two separate processes, but US citizens who legitimately adopted an orphan in a foreign country can obtain US citizenship for the child.

There are 2 components to this:

1) the adoption process

2) the child's immigration

It's much easier to adopt a child in another country, so it's possible for an American to adopt a child there, although in almost ALL cases such American will have also citizenship of that country.

The tricky part is for that adopted child to get an immigrant visa to the United States. For that to happen, the child has to be under 16, an orphan (both parents dead), and no next of kin willing and/or able to care for that child. If so, the child can get an immigrant visa and becomes a Green Card holder upon processing. The next morning the child automatically becomes a US citizen based on the Child Citizenship Act of 2000 and the parents can apply for a US passport. In reality, it's better to wait 'til the Green Card arrives in the mail as it needs to be surrendered.

No.  Citizenship is a separate legal process.

No. There is paperwork to go through.

Yes, but only if they travel to the USA on an immigration visa due to being adopted.  

If a foreign child is legally adopted yes the child becomes a citizen