Hi

First of all yes you are eligible for West Bengal State counselling, it doesn't matter even if you are not domicile of West Bengal

But yes ,

You are eligible only for management quota seats in private medical colleges and not for government medical colleges and Government quota seats in private colleges .

This is because ,

Other students are eligible in West Bengal State counselling but only for management quota seats , you can check the same on the page number 3 of pdf available at:-

https://wbmcc.nic.in/ug/ShowPdf.aspx?Type=B51A60734DA64BE0E618BACBEA2865A8A7DCD669&ID=1CC641954099C249E0E4EF0402DA3FD0364D95F0

Again,

Yes, you can participate in mop up round of West Bengal State counselling,

it doesn't matter at all even if you have not registered for Round 1 and round 2.

This is because,

Fresh registration , fee payment and choice filling is available for those candidates who have not participated in earlier rounds.

You can check the same on the page number 13 of the pdf.

Thank you