ITC eligible

yes you can take ITC of that

Yes definitely you can claim credit of GST charged on the bills.

They need to mention Your GST number for claiming the GST. 

Supplier has raised valid bill with your GST No.

If the supplier uploads the data including your GST no.

The credit gets autopopulated in your Electronci credit register

If not in register- you may upload

In in ECR then verify and if ok - credit is provisional available to be utilised.

The supplier needs to pay his liability and file his valid return.

If he does so - your credit is final. 

If he does not then the credit in your ECR would be debited to your output liability  along with interest for that period and you would have to pay GST.

Between the eligibility and actual final free use of credit a bit of procedure by supplier + you is involved. 

and files valid