Greetings incoming freshers, second semester American here!
It is completely normal for uni assist to take a long time, I actually didn't get my admission until the end of September after i already arrived in Europe. I recommend looking into housing a lot before you come here as I got stuck without permanent housing for a long time. A popular thing Germans will do is choose a flatshare (the student housing is decent, but it is generally nicer to share an apartment with other students). You can check the website WG gesucht for housing in Leipzig. Another option is to go on the Leipzig Erasmus page and look for housing offers. Many international students will be looking for students to fill their vacancies. There is also a whatsapp group for erasmus students, you can ask about it on the facebook page (Links below). I had some Taiwanese friends that were without housing for a month and they had to find people to live with before permanent housing. I also had to stay in a hostel for almost two weeks before I found a WG (and later moved to student housing because its a little cheaper and Philip Rosenthal is at the Physics faculty) . I recommend Meininger if you are looking for a hotel or hostel to stay in when you get here.
If you are worried about your application package, I recommend calling uni assist and ask them what is going on, I called 2 or 3 times and i think this helped with the processing and eventually got everything through.
On public transportation, your tuition is 216 Euro for the first semester and 206 euro for the rest (10 euro for the card) and this will include your transportation (bus, sbann, Strassenbahn, and even includes bike repair shops. You can bring your bike on the sbann always, and on the strassenbahn and buss after 8pm and before 5 or 6am.
I recommend the housing closer to the center for convenience but some of the further out housing is cheeper and newer (thus a little nicer and cleaner).
Just a heads up on the math. It will be very overwhelming as we begin with set theory and then will slowly get into high school calculus, mean while in theoretical physics you will be using multivariable and linear algebra right away hahaha! BUT it is doable because the exams are much easier than the homeworks and the professors are generous about exam admission. Most of the time your whole grade is based on the exam which may sound stressful but its not because that means if you mess up on homework or forget to turn it in, it won't hurt you in the end (as long as your willing to ask the professors for exam admission or extra points)
A warning on learning German, many people stick within the IPSP bubble and don't learn so much German, but if you have a stronger will and go to parties and meet Germans on your own, and have the passion, you can and will learn German quickly. There are also many Spanish, French, and Arabic speakers here so that is kind of cool. The city is full of international students and there are a lot of international parties you can go to to meet foreign students and i think that is special and pretty dope. Just don't get too overwhelmed by the maths and crazy homework, keep calm, go have a beer, and study have study groups and you will be okay.
There are some IPSP and Physik beginning of the semester social events you can go to. it would be wise to ask for advise from furtrher semesters here and learn how to study. You really need to learn to read textbooks and teach yourself as lectures aren't enough, also don't be afraid to ask for help and message people.
Also if you come from somewhere that doesn't have a winter (eg los angeles, taiwan, mexico), it is easy to get depressed by the weather but if you go clubbing its not so bad. There are also lots of opportunities to travel around Germany, Poland, and Czech with Wilma, I recommend this.
I will post the math prof's website so you can see the books we use (she has free downloads) Also in theoretical physics we use Wolfgang Nolting which is pretty overwhelming and dry for first semester, but mechanics is never that interesting. The more interesting stuff will come later (unless you like mechanics, then good for you but imo Electro, thermo, quantum etc are more interesting and you will cover later).
Oh and on age it ranges from 17 - 40, most people are in there 20s though, you will meet people from many places in life. We are also in the middle of finals right now so that is probably why not so many people are responding from higher semesters. I will try and get more people on here and check regularly to answer any questions. I remember last year many people were kind of lost as German Bureaucracy can be complicated. Also a warning, Meldebestätigung is different than Aufenthalterurlaubnis (residence permit), just make sure you have a visa. If you don't have one you could stay here illegally without knowing it and nobody will tell you until you try to leave, i made that mistake when i came here and spring break got a bit complicated.
There is also Barfög scholarship, but is usually hard to qualify.
I hope this answers most of your questions and feel free to ask anything else you want to know as it can be hard to find information, especially up to date information.
WG Housing
https://www.wg-gesucht.de/
Erasmus Group
https://www.facebook.com/groups/exchange.leipzig/
IPSP Class of 2017/18 Group
https://www.facebook.com/groups/108084599889314/
Math Webpage Semester 1
http://www.math.uni-leipzig.de/~bellova/ipsp/