Is there a way I can check dates (going over 2 years back) on Steam Store purchases?

There may be a limit on how far it goes, but my Steam transaction history goes back to December 2009.

To view your own transaction history from the Steam client, click on your account balance in the upper right of the client to bring up the account page. Then select "STORE TRANSACTIONS" and then "SEE ALL" at the bottom recent transaction list.

If you have no account balance, you can also view your history by going to the "Steam" menu in the top left, selecting "Settings", clicking "View Account Details", and then clicking "Purchase History".

The alternative way is to click steam support and click the game that you want to know the purchase date. Then, look at the top right corner, which will show the purchase date of the game. It should look like this:

You can also just navigate to your game library and see the date each game was purchased (right beside "Install"):