If I download a shared Playstation game and then buy it, will I keep all the progress

Unfortunately, you can't do that directly off the PlayStation Network Store, since the buy option won't be available for games you own through PS+, only after your subscription has expired will that option be available again.

However, you may be able to purchase the full games through third party sources. For example, GameStop offers download codes for PSN games that can be redeemed on the PSN Store. I assume that if you redeem a code for a game you only "own" through PS+, you'll own it permanently (you can check the game's information in the XMB and see if it has an expiration date).

I cannot confirm this method 100% as I haven't tested it myself, so try at your own risk.

You don't need to physically download the games to your device, but you do need to go into the storefront (either through an appropriate console or through the web interface) and "purchase" the items by adding them to your cart and going through the checkout process. Once you complete checkout, those items will be added to your Download List permanently and can be re-downloaded as many times as you want, even if those items are no longer free for new PS Plus subscribers. You will lose the ability to download or play those games if your PS Plus subscription lapses, but if you subsequently re-subscribe, you will once again be able to download and play those items.

This does mean that free games can potentially be "missable" if you have not added those items to your account before they go back to paid status. While some Instant Game Collection games have remained free for long periods of time, that is often not the case, and you will need to be sure to "purchase" those items during the limited window in which they're available for free.