Bike for many short city commute AND long distance travels (partially dirt road)
If you want to see if a bike can take a luggage rack, you should look for eyelets for a rack. These are on a lot more bikes than just city bikes.

If you're looking for a rear rack mount, near the dropouts, you should see a set of holes or two, and a set of holes on the seat stays so that you can attach a rack. For a front rack, you should have eyelets at the bottom of the fork, and some bikes will have another set half way up the fork (though some racks will attach to a hole at the top of the fork + some p-clamps).

It's less ideal without the eyelets for a rack to install a rack -- normally you have to do some work with fidgeting with p-clamps and stuff, so its not recommended (or use a seatpost rack, ick!). 

Installing a rack when you have eyelets is pretty simple -- you generally bend a small piece of metal to make the rack to be level, and put a few bolts in. 

Most hybrids have rack eyelets, as do many cyclocross bikes, pretty much all touring bikes, a decent amount of road bikes, a good amount of lower end mountain bikes and so on. They may not ship with the rack like a lot of city bikes or whatever, but its not hard to put one on. You can easily look for the eyelets or just ask the person whose selling you the bike if it has rack eyelets.

This article has an overview of different common rack types, which shows you how they mount. It's relatively obvious though -- just walk into a bike shop with a bike, and in about 5 seconds they should be able to show you what kind of racks will fit it.




  This bike is not designed for cycling on rough terrain.


Generally, a statement like this means you don't want to do serious off-roading with it. In the US, there's a set of condition numbers from the ASTM:


  
  Condition 0 — Adult supervision required, no traffic;
  Condition 1 — Suitable for road riding (only);
  Condition 2 — For off-road riding and jumps less than 12 in. (30 cm.);
  Condition 3 — For rough off-road riding and jumps less than 24 in. (61 cm.); and
  Condition 4 — For extreme off-road riding.
  


The bike under consideration is something like a condition 1.5 bike. You don't want it to jump more than say, 6 inches. See this chart from Salsa for more examples. But if you're just riding on fire roads or the gravely dirt roads you show above, it'll be fine.



Now, onto the type of bike you should get. I'd consider a cyclocross bike with good tire clearance, a touring bike with good tire clearance or a hybrid with good tire clearance. An older (say 80s) road bike would probably do fine too, since they could take bigger tires). The touring bike might feel weird when not being loaded, but the advantage of the first two is that they have more hand positions available for a longer ride (and you can install interrupter levers to stop from the flat portion of the bar, if you want). Larger tires = more comfort and cushioning. 

We don't do product rec for specific models here, so thats about as far as I'll go in this answer. But, I'd probably go for a cyclocross bike myself. Theres a pretty wide variety in this range -- things like the Trek Crossrip, which take a cross geometry and tweak it for more commuting/city use, to bikes which are suitable for intro level racing yet can still take a rack and fenders (like the Kona Jake the Snake used to be until a few years ago). 

I'd also remark that you don't need (or want) suspension for that type of riding. 



The most important thing for comfort is to actually try the bikes -- don't just buy something from the internet (especially since sizing isn't standardized across bike types).