What would a typical user use such a network for?
Well we talking about interplanetary internet so the user will (want) to use it for what we use it today.
* Communication
* Information
* Entertainment
How reliably would it work?
I would assume that the network would be very reliably (on the higher layer). The question would not be if the message arrives, but when. However interference could delay a message for hours (or days?).
What would the expected bandwidths be?
I think we can build a network that has enough bandwidth, but I didn't found information on this. So the rest of my answer assumes we have enough bandwidth.
Would using it be exorbitantly expensive?
I think the network would be quite expensive. However not necessary for the end user. A system wide internet could be subsidized by government and content provider which want to sell there content to the users. That's what happens today in not so dense populated regions to get fast internet there.
And how would it actually feel to use such a system?
If the user access planetary (his current location) resources there won't be a difference from today. As soon as he actually uses the interplanetary network, however somethings most likely will change drastic.
Entertainment
Any online game that has real-time interaction won't work. You could still have something like racing against ghosts of other players, simple leaderboards or playing something like chess. Like in the time when you played chess via mail (not E-Mail).
Instant Video wouldn't be so instant as long as it is not cached anywhere. Big players could maybe hold a copy of all there content in different datacenters around the system. However if you want to use a service that don't provide this, the question would not be "what should we watch (now)", but "what should we watch tomorrow". So you will request the content and watch it when it comes. If you have an abo for series they will start sending you a copy to watch offline as soon as it is released (maybe still DRM protected).
Information
News will be as today. You will still be notified but it will take longer. This could result in a trend back to traditional media. If you need to wait several hours for your information maybe you can wait an hour longer for verified information.
Surfing the internet from link to link will not work. There could be an indicator how long it would take to load a Link. 205
Is it a local resource or it is already cached. Often accessed sides will (as already today) be cached. But good caching will get much more important. Surfing will often result in I want to read this and that later. On the technical site requesting a Website would not only deliver the site but also other sites of the domain and other linked content directly.
Communication
We would not use messengers as much. There wont be any direct communication. We will go back to E-Mail, or something similar. Maybe we would write more and longer letters, like in the time where there was no telephone. Communication on websites should change to (in order to be still somehow use full).
Forums will need to structure the communication better. You need to know which post related to which other in order to follow the discussion. Some websites will only used locally because the discussions are to fast to allow anyone from far away to participate. Other Sites could use some mechanism to counter this by showing posts only after a specific time they where posted (when the user pressed send, not the server received the post).
I would assume a raising popularity of IaaS provider that will replicate your content over the howl system. Even small blogs that currently are hosted in private environments maybe want to migrate to bigger providers. We would maybe also see more toplevel domains that indicate the location, like .us.earth or .mars, .pluto and .sun.
The underling protocols of the internet must change drastic as the application the end user uses must change in order to give him enough information what will happens if he clicks this link. Nobody wants to click a link in twitter and sitting there for hours until the site loads.
I think log distance communication will feel more like in old days before telephone. There wont be any non local news with live broadcast from the place where something happens. Nobody can ask questions to the journalists at the place. Most likely the information will still be published as soon as it is available. But the livetcker would not be live anymore. 897