1) Loyalty cards - you can find all the information on the Internet, or just ask in the supermarket. As for coupons - we don't have many of those going around, sadly.
2) Supermakets - as people mentioned, Biedronka is the best. Lidl is also ok, although in my experience I pay more money there for similar products. But special offers are a thing, so you should keep an eye for them and make your rounds, if you have the time. If not, Biedronkas are there for you.
3) Food - really depends on where you're going to live exactly, but in general chain restaurants will be your best bet. Still, you are going to spend at least 20 - 25PLN for a meal in those. If you want cheaper food, kebabs and similar places can feed you for less. If you really want to minimize spending on food though, buying ingredients and cooking for yourself is your best choice.
4) Drinks - depending on the location you can expect a beer to cost you anything from 5 to 10PLN - and we are talking about the same brand, just different bars. It is generally cheaper to just buy a beer in a shop - then, unless you go for something fancy, no matter where you're buying it shouldn't cost more than 5PLN. Side note: if you buy a beer in a shop and just want to drink it somewhere outside with friends, you should know that drinking on the street (except places like outside tables belonging to a restaurant) and in parks is illegal in Poland and you can get a ticket for that. However, even if you are not on the street or in a park (for example, you are sitting on a beach on Wisła's bank) and police come, thay may want to ticket you for drinking in a public place anyway - you can argue with them, since it wouldn't be illegal to do that, or just keep your eyes open and hide your bevarage if needed, your choice ;)
6) Earning money on the side - if you speak ANY foreign language on a decent level, you can give private lessons, native speakers naturally have it easier. You can post your offer or search for pupils on gumtree. Just keep in mind, that summer vacation lasts from the end of june to the beginning of september, so there won't be much demand for tutors in July and August. I would say that for a native speaker giving lessons in Warsaw, the rate would oscillate around 50PLN for 60 minutes.
800 PLN is enough to survive without problems in Warsaw, and you should easily have some leftovers for something else than bare necessities. Good luck on your internship!