If you are looking on line I cannot understand why it is so expensive for a short term let. You will get a family room in a cheap hotel for less.

Check the prices. They are sometimes in local currency, but online they use the '$' sign as not all websites read Chinese character for currency.

There is a guy who often advertises on here for apartment letting. Do a search.

Er Hai is big, and there are many sites you can visit in the area, they are not all the same price.
You will only pay the conservation fee if you go into conservation areas.

Some sites, like Xi Hu, will cost over 100.

As you say, the whole thing is a bit complex, but I wouldn't sweat most fees as they are quite low.

I would avoid the package trips, usually offered by hotels and guest houses, as you lose your independence. There is plenty to do within walking distance (5km) of the old town.

I think the same was as @Tonyaod.

It is a customary comment, like 'How are you?'. Generally the asker is just performing the ritual. They don't really want to know if you have eaten. To reply 'no' actually presents a problem for the asker, what are they then supposed to do? Buy you lunch?

My answer is usually 'Che fan le' or 'Che le'.

My thoughts

In answer to your main question, the schools here are very flexible and anything is possible - they want your money.

If you are looking for funding from your own government, they may require that you go to a public sector school/university. Only your own government can answer this question.

What will you be studying? Is the next question.
If it is Chinese, and you are a true beginner, unless you are truly gifted in languages you would most probably not be able to catch up. There are no links with latin based languages, and as such there will be few areas of similarity with your mother tongue to draw upon (unless you know Japanese). The universities study with a 'Chinese' learning pace and stlyle, which often means lots of night study to keep up, this is a reason why catch up is so difficult.

Don't just wait. Even DHL need chasing here sometimes.
If the parcel does not have your phone number on it, you will need to go to the post office to chase them.
Expect for PO staff not to be able to find parcels the first time.
Don't go just before lunch, you will get fobbed off.

Take cash as you may have to pay customs duty.
Expect to have to provide an inventory/packing list, if it did not come with one.