I`m assuming from your name that we are from the same country,am I right?
I also assume that you are still in you home country.
Answers to your questions:
1. Yes, as a PR your girlfriend can sponsor you- in the following case: you are married and you apply under spousal category. If you are not married you can apply as common-law partners if you can prove at least a year of cohabitation. Your girlfriend does not need to prove income to be able to sponsor you.
2. If you want to come to Canada and be with her, you can as a visitor. Hungary (correct me if I`m wrong and you are from somewhere else) is visa-exempt, means you can be here for 6 months without a visa, but you have to state at the entry that you have financial support for the time you spend here.
3. Getting a work permit is possible during your PR application is in process once you get AIP ( first stage approval)- if you apply inland it takes about 9 months to get it. You have to include an open work permit application with the PR and pay the fees as well. It will allow you to work for any employer once you get it within the timeframe CIC gives you.
4. In your place I would consider whether inland or outland application is better. Outland is way faster and you can still be in Canada as a visitor while it is in process. Inland takes about 18 months total, you have the advantage though to live with your girlfriend and the right to remain here until decesion will be made in your pr appl.
5. If you apply inland it is not recommended to leave the country as there is no guarantee they will let you back in and it would cause you appl. being rejected. I would not risk to leave, you never know. Your girlfriend can leave the country though- not for years.
6. If you submit your application you need a lot of proof of your relationship being genuine. Means you need joint account statement if you have, residential lease with both of your names or home ownership, utility bills with both names, marriage certificate, passport on your side, PR card on her side, will/life insurance stating each other as beneficiary, any online chat, conversation, mailing with each other from the begining of the relationship, photos of both of you with families on outings through your relationship, any latter from friends, family members confirming your relationship, any receipts, tickets from trips, vacations together, birthday/christmas,ect. cards titeled to each other, your police clearance from your home country, your medical...and anything else that could help your application.
It is not an easy process, especially the part to build enough evidence- it takes months for most people. You should look up all the information about PR application and requirements on CIC website.