How big of a deal is it to miss the deadline for renewal of a third class medical? Is it simply flying privileges are suspended until it is completed, or does it involve a more extensive renewal process than normal?
Nope, the medical process is the same whether you renew it early, in the last month, or ten years after the last one.
As long as you don't act in situations as a required crew (no being PIC, safety pilot, ...) you can get by without it. Notably, you can receive flight instruction from an instructor qualified to be PIC (better verify that if you do).
Right -- it's not like renewing an expired CFI ticket where you have to do a lot more if you go past the expiration date. The exam for a medical is the same whether you apply for a new the day you got the old one or ten years after your last one expired. Just don't do anything requiring a valid medical in the interim.