The best way is apply thailand tourist visa in laos vientiane. You can enter from border with visa only. Arrival visa is not allow from borders. Its only cost $40 and about 3 days
I think u may have an interview email soon. Ur interview date may monday. If u can show ur bank account from ur mobile phone at interview and u have good money evidence u may get visa on monday
sponsorship letter is about u have to pay all the cost of ur childrens. And family certificate is the list of ur family members approved by goverment(some other country may different).
Process is u have to apply to get visa first. If u have visa. U can apply dtv dependent visa for ur children. U need birth certificate of them and family certificate and a sponsorship letter for them and bank statement