Andrew Jackson is known to history as an impulsive, reckless guy who took personal slights and adopted governmental policies in reaction.

There are two things in your question. Was Jackson an impulsive, reckless guy who took personal sligthts ? Yes. As multiple biographies (references below) suggest it, he had a big temper and could hold a grudge for quite a while. It's a trait he demonstrated when he started to make a name for himself as a lawyer, and it followed him all the way to the White House.

Now did he adopted gov. policies purely in reaction to personal slights and reckless atitudes ? It's more nuanced. If you take what are probably his two most famous policies, you can track back an ideological and sociological background for each of them that is, arguably, more influencial that personal slight. I'm talking his Indian policy and his refusal to renew the charter of the BUS.

Yes, there are some aspects (especially concerning the BUS) where personal grudges are relevant to understand his stances. But there are bigger factors in his past that allow us to understand his positions in a more refined way. On this precise topic, I can only recommend Avenging the People, by Jason Opal.

On the actual questions, yes it was know while he was in office. I'm not sure we can say it was known by the people, but for sure it was known by his enemies. Most of them have had to deal with him for quite a while, whether as a US General or Congressman, and they knew about his temper. They knew so well about it in fact, that a whole part of their strategy during the 1828 campaign was to try to provoke him and make him lose his temper, most notably by questioning his military record and namecalling his wife Rachel.

The second question is harder to answer. We didn't have any modern poll to question the population about such things. The fact is that he won 1828 and 1832 Presidential elections quite handily, and his military achievements probably overrid his personality traits.

Sources :

Avenging the People, Jason Opal (2017)

The election of Andrew Jackson, Robert Remini (1963)

What Hath God Wrought, David Howe (2007) (not focused on Jackson per se, but speaks lenght about the jacksonian period)

Andrew Jackson: His Life and Times, W. H. Brands (2005)