When buying a ticket without a reservation on an Intercités train, is there any advantage to buying it in advance, instead of just before departure?

SNCF uses Revenue Management techniques on the pricing of Intercités. So, yes to your first question, the price varies over time and most likely, the earlier you buy your ticket, the better deal you'll get (from my experience this is almost always true). But for this, you will need to make a firm reservation and will not get a refund on the exchange/cancellation of the ticket.

In case you want flexible conditions (it sounds so), the price is always the same, whenever you buy it. You can exchange/cancel your ticket before the train departure with a 10% fee if you do not make a reservation. The ticket is valid for 60 days from the day you bought it.

As for your second question, I never saw that Intercités had no more spot on the train (but I rarely take these trains), so it could probably happen and I would say it is more likely to happen on Friday evenings, Sunday evenings or even weekday rush hours, depending on the area (Intercités to Paris might be used as commuter trains, someone else could confirm that).