You can make the appointment before your arrival though, in order that the process does not stretch too long into your stay.
You will also have a health card and a cedula from very shortly after this appointment. So only benefits to making the appoint for shortly after your expected arrival.
Yes, you arrive on a tourist visa for 3 month extendable to 6.
You can apply for residency whilst in the country but remember some of the document requirements are apostilled Birth/Marriage/financial report/criminal record checks from your origin country, which can take some months to arrange.
I am currently going through the process arrived in DEC
Yes, if you apply from abroad, you get a QR code that is needed to pay in cash at a local Abitab branch, so for remote option you would need a friend already in Uruguay or a agent to physically pay it for you. There is no option to pay by bank card.
Articles to help you in your °µÍø½ûÇø project in Uruguay