It's really tough to answer. I've been on a ride several times with 5 hills that MapMyRide classifies as category 5. Most of them aren't bad at all but two are pretty challenging (for me). Race officials generally classify those 2 hills as category 4. There are people that go on that ride with just single speeds and do fine, but they're strong riders.

It's probably safe to say that these are hills on your route are ones you're at least going to notice.

Given that, I don't think a 3 speed is the best choice but it's really a matter of how hard you want to work on those hills.

I think IGHs shine when you're riding in adverse conditions like lots of grit or road salt. I don't think rain in and of itself is so hard on derailleurs. There other advantages to an IGH. You can shift while stopped, and you can use a full chain guard. Chains are cheaper and will last longer. IGHs are less likely to get knocked out of whack by your bike falling over (or other accident).

Downsides to an IGH: Upfront cost, weight, a bit more difficult to fix flats, repairs can be very expensive.

Do you need more gears than 8? I can't say for sure. Even with 8 gears the shop can change some things to make it easier to go up hills at the expense of maximum speed. For example, I have an 8 speed bike an IGH that has a lower "Low Gear" than my 18 speed road bike. All else being equal, it would theoretically be easier to get the IGH bike up a steep hill than the road bike but I wouldn't be able to get it going as fast down the other side of the hill.