The way to work it out for yourself is to think about it this way. Imagine a "normal" customer on a one-night stay who, say, checks in at 5pm and checks out at 10am the following morning. Any time you're in your room, that other person can't be there. The other person would have obviously paid for their night's stay so, unless the hotel gives you a nice deal, you must pay for that night, too.
When you arrive at 2:30am on Monday, the hotel had to keep your room empty from 5pm on Sunday, so they couldn't accommodate one "normal" customer. This means you need to pay for the whole night of Sunday-Monday: you need to book the hotel as if you were checking in on Sunday.
Likewise, if you leave your room at, say, 11:30pm on Thursday to catch your flight early on Friday morning, the hotel wasn't able to accommodate a "normal" customer checking in at 5pm on Thursday, so you would need to pay for that night, too. However, you can get around this by checking out of your room earlier. Some hotels will allow you to check out a couple of hours later if you ask them; some may require a small payment for this (much less than the cost of a night's stay). Most hotels in my experience will look after your bags for you after you've checked out. I've never had to pay for this but apparently some hotels do charge.
So, my recommendation is this:
- Ask your hotel if they can look after your bags between the time you check out and the time you leave for your flight.
- If they can do that, book the hotel for check-in on 12th February and check-out on 16th February.
- If they can't, you'll have to decide whether you want to pay for an extra night (check-out on 17th February) so you can leave your bags in your room, or if you'll check out on 16th February and look after your bags yourself for the day.
- In either case, advise the hotel that you will be arriving very late, and estimate what time that will be. This is important because they might cancel your booking if they think you're not coming. Most online hotel booking systems have a box in which you can add information like this; if you're booking over the phone, obviously tell them when you book.
- Consider late check-out on 16th February, if it's an option and it's reasonably priced. That would give you your room until around lunch time. (If the hotel will look after your bags, this probably doesn't matter much to you.)