For the bank account, you should decide if you want someone to talk to. A lot of people end up with Deutsche Bank initially, because they needed a special type of account to get their visa that only Deutsche Bank provides. But they are also expensive later on. The normal bank account you need for your salary is called Girokonto. There are banks that provide it for free and have physical branches, like Commerzbank. There are also online-only banks that do it for free. Some banks with branches charge up to 10 Euros a month for the account. It's up to you what you want.
With your account you will get a Maestro debit card that works for paying with a PIN and taking money out of the ATM. Most banks charge a fee for using ATMs of different banks unless they are part of the same group (cashgroup is one of them, where Commerzbank, Deutsche Bank and others are part of it). The Sparkasse is something similar to a bank, with branches, and lots of ATMs everywhere, but they are limited to the city where you are. With an account at Berliner Sparkasse you will be able to use an ATM in the Sparkasse in Hamburg (Haspa) for free to take money out, but not go to the cashier in the branch.
Bank accounts do not typically include credit cards, but there are some that provide a free credit card too. Usually you have to pay a yearly fee for credit cards. The acceptance for those is limited. If you do not want to pay cash, use the maestro card. And remember that change is quite valuable here, unlike in the US.
I don't know how hard it is to get an account in an online-only bank. There are types of work visas like the blue card where you need to get the bank account fast. It's a bit hard to get started, especially if you don't have an address yet, and a bank with a branch that you can go to, and maybe take a new coworker with you to help with paperwork goes a long way.
With regards to 3, it depends. In Germany we usually distinguish between so-called contracts and prepaid top-ups. Decide what you need the phone for. Do you want to do a lot of calling to other countries over the phone network? Do you need a lot of mobile data? Do you want to send SMS?
At the moment you can get decent rates with free unlimited calls and SMS pretty much on every network. Those are called Flatrate or Flat in German. Both the large traditional companies as well as cheap, mostly migrant-targeting prepaid providers have those and you can get a decent deal for 10 to 15 Euros a month easily.
Some of the large, traditional providers include T-Mobile, O2 and Vodafone. All of them also have prepaid, in which case the network coverage is mostly better than with the prepaid-only companies. Almost every discount supermarket chain offers prepaid, but the prices vary. Some of them are Lidl and ALDI. There are also prepaid-only companies that are ventures of the traditional ones, like Klarmobil or Congstar. Then there are the ones that target migrants and low-income people, like Lycamobile, and Lebara as well as maybe Turkcell. They offer special packages you can book after you've topped up to call certain countries for free or very cheap. There are deals for Turkey, the Middle East and northern Africa mostly. You can buy top up for those things in every little Spätie store in Berlin and you can top up as little as 2.5 Euros.
Typically mobile data comes in packages of 300MB, 1GB, 3GB and 5GB and cost somewhere between 10 and 20 Euros per month. For the prepaid things, you typically top up, then send an SMS to activate one of the packages, which will then last for 30 days and might automatically re-book if you have enough credit.
If you intend to stay long and you have a secure job you can make a contract with a traditional carrier, but keep in mind that those usually come with a 2 year attachment, and the only way to get out of them early is to prove that you are moving away from Germany. If you're not sure, get prepaid with a large company, then later move your number over to a contract if you want.
If you intend to travel within Germany the whole data roaming should not be an issue with any of the providers any more starting from the middle of June this year, as a new law will basically remove the costs and you will be able to use your data anywhere in Europe without extra charge (hopefully).
In addition to that, if you want landline internet at home, you can either get DSL or cable (Kabel). Again most companies will only make contracts that have a minimum duration of 24 months and you will have to pay a fee to change your address if you move around. You will also get a landline phone number, but you don't need to get an actual phone and connect it.