In my case the stretch will be between Bologna and Rimini in the summer.
Are reservations required? Or is it realistic to show up at the train station and get tickets / seats on the next train?
According to The man in seat 61 regional services do not need a reservation, whereas, other services do. Some services between Bologna and Rimini are regional services, others are not.
To find out which is which you can use SBB, enter your arrival and departure station (choose the exact station in Bologna you want, centrale is the main station but others may be better for you) and select the exact date. If you follow through the form and download the timetable, to work out what type of train look in the journey collum on the left. Services with an R are regional services with no reservation required, other services require a reservation (marked ES or IC), these do though tend to be faster.
You can also use this website to find the type of train being operated. Again with R meaning regonal train where no reservation is required. In adition, if you click on a train and the phrase Prenotazione obbligatoria appears under the Servizi e Info section then a reservation is required for that train.
Yes, you need reservation on some trains.
But what is reservation, for the Italian train service? Simply a mandatory raise in the ticket that you didn't request: as reserving a seat was a luxury option, making it mandatory means that everybody has to pay it, thus the friends of politicians makes much more money.
Now, as it is included in the ticket, for you it makes not much of a difference if a train has reservation or not. You buy your ticket at the train station, 30 seconds before the train leave, and you pay a total which include reservation. You buy the ticket one week in advance online: you pay the same amount. Additionally there is a small trick: what is mandatory is not to have a reserved seat, but to pay for it. If there are no more seats available they will usually sell you the ticket anyway (at full price, obviously) and you'll just travel standing, so you'll still be able to take the train.
My advice: buy the tickets in advance in the morning when you arrive at your destination, so you are ok to go when you go back. Easier, cleaner, less worries.