The first document is the provisional cédula de identidad, issued once the residency process is open. It is what unlocks a bank account, a health plan and a rental contract.
Legal residency and tax residency are different things and do not happen on the same day. You can be a legal resident without being a tax resident, and that distinction defines where you pay tax.
Uruguayan state schools are free and accept foreigners before residency is finalised. Bilingual schools in Montevideo and Punta del Este have waiting lists — it is worth writing months in advance.