only those I know/remember (I lived in Poznan for many years, but moved out 5 yrs ago):
one remark first:
generally, if you're not the owner of the appartment you live in, most companies require the owner to sign the documents/contracts for you
1/ What is the name of the local internet company? For Cable
Telekominikacja Polska SA (TP SA) - national phones and internet company (Neostrada is their most popular offer for ADSL - worth it when signing a 2-3 yrs contract, you pay about 50-100 PLN/month, otherwise it's usually quite expensive), Cable Internet was widespread in many areas, but there were several different operators, depending on where you live, you have to contact your local Cable TV for that, if yo don't have cable TV in your apt., installation was be a little expensive, maybe it changed.
2/ What is the name of the Electric Company?
Zakład Energetyczny ...
3/ What is the name of the Heating/Gas Company?
heating in blocks of flats is usually included in the rent for the apt., you pay all year round (even in summer, but thanks to that in winter you pay less than the real costs)
I don't know the names of the companies, maybe Zakład Ciepłowniczy (heating), Zakład Gazowniczy (a wild guess here, we usually say "gazownia", it's a colloquial term)
4/ What is the name of the Water company?
5/ What is the name of the place to buy a Student Tram Pass?
the bus/tram company was called MPK, look for the logo at small "kiosks" at end of the line, or for example in Rondo Kaponiera (in the centre near Bałtyk cinema)
the monthly pass is usually called "sieciówka" (pronounced 'shechufka" actually it's not "sh" and "ch" but softer sounds, if you know a little German, their "ich" is pronounced in some regions almost as softly as Polish ć/ci)
there were also 24 hours tickets and 6/12 months passes a while ago
normalny/normalna - means full price
ulgowy - with reduction (for students for example, you need to show your student ID when buying o be sure you're entitled to a reduced price, also keep the student ID with you when on tram/bus)
6/ What are the rules for tipping waiter/waitresses? Is it
included in VAT? (like in Germany)
no rules, I think, never heard of it included in VAT, most people tip (5-10%), some don't.
7/ What is the best place to buy a new or used Oven?
+ 8/ What is the best place to buy a new or used Washing Machine
Used - I don't know, new - there are many shops, usually labelled AGD (artykuły gospodarstwa domowego), often with RTV shops (TVs, DVD players, mp3 etc.), so look for a AGD or RTV - AGD and check prices
9/ What is the best place to buy small passport sized photos,
to be used for school ID and such.
just a regular Zakład Fotograficzny, they usually make it on digital cameras and print within a few minutes, in Poland there are different sizes of photes required for different documents, but the people at Zakład Fotograficzny usually know the rules, so you have to tell them for what you need the photos and you should be OK.
10/ What is the best Bank for students? Do any offer Free
checking, online access, or free ATM fees?
ATMs are usually free if you take the cash at an ATM belonging to your bank, local Poznań bank is WBK Bank Zachodni (or something like that), so they have most ATMs in the city I guess, but I can't tell you about their offers, just visit a few banks and check where people speak good English and ask them for help :)
Many years ago my University had a contract signed with this bank, so when you opened an account there through your Uni, you got better deal than in other banks, not sure if it still exists
Online access is usually not a problem, but the Polish banking system is much of a mess, one bank lures you free online access, other with free money transfers, another with the highest number of free ATMs in whole country etc., so you really need to find an offer that suits most your needs
11/ How does mail work in Poland?
What is the Post Office called?
Urząd Pocztowy (colloquially simply "Poczta")
Do I just order items to be shipped to my doorstep address, or do I need a PO Box?
your house/apt. should have a letter box, if you rent the apt., the owner should give you a key to it, normal letters are left there by the postman, registered mail or packages - you must receive in person (or someone leaving in your apt. can do it for you), if you're not at home, the postman leaves "awizo" in your letter box, there you can read what type of mail they got for you, where you can receive it (it's Urząd Pocztowy with number and address), what are their office hours etc.
12/ Is tap-water drinkable?
not poisonous, but not recommended in many areas, I hope things improved since I moved out, but like someone said a filter is a good idea
13/ I have a bicycle. Can I just lock it anywhere on the
street when I goto school or the market?
not on the street, but on the pavement (sidewalk) :)
there are often special stands for bicycles, but if you go to a small shop, then usually you don't find them, and don't forget to lock the chain (or whatever type of locks you'll be using) if you have a nice bike.
BTW, you can't take your bike on tram/bus and on train you must buy additional ticket for it (very few trains that allow the bike without paymnet) and most probably put it in a special luggage car (called "wagon bagażowy")