A shamisen of the beginner set is one piece instrument ie there is a one piece neck (no a two part neck) fixed in the body? Thanks.

Special Set and Beginner Set are nobezao type, which is impossible to separate neck and body. 
Others are mitsuorisao type, that can be separated with three parts. 

Hello! Does the Special Set come with accessories?

Accessories of special set:
Neck cover 天神袋
Slip guard 胴掛
String for slip guard 胴掛ひも
Plectrum バチ（ネリ）サヤ付き
Finger suck 指掛
Rubber 張りゴム
Bridge 駒
Bridge case 駒入れ
Additional string 予備糸（ナイロン3）1本
Textbook (in Japanese)入門書

Is this an ST2 or ST3?

Beginner set is ST2 and popular set is ST3. 

Thank you.

What all is included with the beginners set

Accessories of beginner set: 
Neck cover 天神袋 
Slip guard 胴掛 
String for slip guard 胴掛ひも 
Plectrum バチ（樹脂製）サヤ付き 
Finger suck 指掛 
Rubber 張りゴム 
Bridge 駒 
Bridge case 駒入れ 
An additional string 予備糸（３の糸ナイロン）1本 
Textbook (in Japanese)入門書

I can't find beginners set anymore. Did the special set replace it?

The maker discontinued the beginner set and it's unavailable anymore. We may ask to make azuma sawari part on the special set but it costs money and takes time. Is it ok?

does intermeditate set come with case?

Yes. The intermediate set comes with a case.

Which set has the aszuma sawari

Beginner Set, Popular Set, Intermediate Set, and Advanced Set have the azuma sawari. 

Does the Beginner set come with a case to carry or must buy seperately?

Beginner set doesn't include a carrying case. Hard case and soft case are sold separately.
https://taiko-shop.com/collections/japanese-instruments/Shamisen