Don't begin eating until your host says "do brou chut." If drinks are accompanying dinner, the proper response to a toast is "naz dravie" (to your health). While it is customary to turn down an offering of more food, this is again just a formality; your host will soon be asking you again if you'd like seconds and only upon their second attempt should you accept.
If you are finished with your drink, just leave the glass more than half full to indicate you are finished and at the end of the meal (if at a restaurant) be sure to offer to pay. Again, this is just a formality and your host will thank you for the offer, but will almost certainly turn you down. If you are the host, do the same by paying for everyone, but thanking them for their offers to contribute to the bill.
When eating at a sit down restaurant, in regards to tipping, rounding up is appropriate and all the locals will give nothing more than this, however at restaurants catering to tourists, a tip of about 5-10% is expected.
Drinks
Slovakians enjoy their coffee, but mineral water, soft drinks, and other carbonated drinks are king in Slovakia when it comes to non-alcoholic beverages. If seeking out something more familiar, Slovakia also has juices, tea, coffee, milk, and international soft drinks widely available.
Slovakia produces their share of alcoholic drinks; two local specialties are slivovica, which is a fruity plum liquor and borovicka, which is also sweet, but distilled from juniper berries. At pubs or restaurants however, beer and wine are much more popular, although hard liquors are also widely available, all of which include international brands.
Generally speaking, the tap water is safe to drink in Slovakia, but check with locals for any particular regional differences. Also, many people may have troubles adjusting to the local tap water, as it will most certainly be different from what your system is used to.