I will answer the second part of your question.

did any areas recover faster than the rest of the country?

Yes. It is important to understand that while the US as a whole was affected severely by the Great Depression, the impact of the crash varied from State to State and the three primary factors were the extent of the sectoral specialization, levels of the household debt of a particular State and spending by the Federal Government during the New Deal. For example, Agriculture, Banking, Mining, and Manufacturing were hit in a far harder fashion than Transportation and Service sectors. In general, the more specialized a particular region was, the more vulnerable it was towards external shocks and slumps in trade both internal as well as external. Similarly, high levels of debt that were accumulated during boom time forced a very painful deleveraging process that necessarily cut down Investment, Consumption and consequently Output. The third factor was the response by the Government in response to the crisis. Another component was the Banking policy which limited the size of the banks by imposing restrictions on expansion forcing most banks to remain relatively small and undiversified. This led to bank failures and worsened the situation further by reducing the confidence of the public in the Banking system as a whole until the whole sector was restructured (I have described that along with other policies that were implemented in detail here.

Research by Schwartz and Graham, Fishback, Wallis have found that States with lower household debt performed better than States with High Household debt when it comes Employment, Income, and Wages and Retail Sales (Retail Sales is considered to be a very useful metric insofar as the measurement of Consumption of both durable and non-durable goods are concerned and it is still used by the US Bureau of Labour Statistics). An increase of 10 percent in debt to income ratio above the national average resulted in the above factors being 2.5 to 3 percentage points lower. Now we come to sectoral specialization. Having a relatively low debt by itself did not save any State from the downturn. States on the East Coast had higher debt to income ratios than the ones in Mountain State and yet the latter had to bear a greater brunt owing to their dependence on Mining and Lumber Industry and less diversification. The collapse in Construction Industry created considerable difficulties because they could no longer find a buyer for the product. Similarly, States that derived a greater share of income from Agriculture fared far worse than others. Lower income States tended to suffer worse in these conditions. The relationship is less clear with regards to Manufacturing. On the other hand, the States that saw a greater fall in income between 1929 and 1932 because of their sectoral specialization also had a faster recovery in the period between 1932 and 1940. Average per capita income in 1929 was $845 in the Northeast, $631 in the Midwest, $661 in the West and $499 in the South

Now, we come to State spending during the period of New Deal. There was (as expected) considerable variation on help provided and the impact it had on the economic performance. The Federal Government spending increased from 4 percent in 1932 to 8 percent of GDP in 1939. Similarly, the Federal share of total expenditure increased from 30 percent to 47 percent. The per capita amount received by South was less than West though greater than North East. Generally, the expenditure on public works was greater in those areas where there were more unemployment and spending on relief was greater where there was more distress. It must also be noted that was an arguable bias toward those areas where there was a greater volatility in support for Democrats. Nonetheless, the impact was mostly positive and the greater income allowed workers to spend on buying consumer goods allowing the economy to steadily recover. The increase in productivity cut transportation costs increasing the net price local sellers received. The efficiency was greater in those cases where spending was not solely used to reduce unemployment and substitute Capital by Labour. Moreover, the Federal spending was instrumental in equalizing and leading to greater convergence in growth rates between States. Federal expenditure was also important owing to the legal restraint on deficit spending by State and Local Governments which might have inhibited recovery. While positive on a whole, there may have been some crowding out of private Investment as it's share was considerably lower in 1940 when US real GDP reached the 1929 level.

In the final analysis, there was variation in recovery rate between different areas based on the impact of the Great Depression and other factors which I mentioned above.

References:

Household Debt and Economic Recovery: Evidence from the U.S. Great Depression by Katharina Gärtner; European Historical Economics Society Working Papers in Economic HIstory | NO. 36

Why Did Income Growth Vary Across States During the Great Depression? by Thomas A. Garrett and David C. Wheelock; Social Science Research Network

The Impact of New Deal Expenditure on Local Economic Activity: An Examination of Retail Sales, 1929-1939 by Price Fishback, William Horace,Shawn Kantor; National Bureau of Economic Research