Pretty girl!!
Yes - food motivated (but she looks fit in the picture)
Yes - runners (they don't intend on running away, they just keep following their noses and oops - they've trotted 3 miles!!)
A few other things about Beagles.
They are not known to be "clean" dogs (none of the scent hounds are), which means they are typically harder to house break, and if they find something smelly - they love to roll in it. These are both instinctive behaviors as scent hounds do not want to be detected by their own scent, they typically prefer to smell like anything other than a dog.
Also, make sure you keep her ears clean - often. Hound ears are very soft and supple (and long) and because they are soft they "mold" into the shape of the ear to almost fill the ear cavity. Because of this, air does not circulate well and they are prone to yeast infections. Clean them often, and Panalog helps (you can get that at your vet).
Lastly - Beagles come in two sizes. 13" and 15" - your's appears to be the 15" variety.
Best of luck! She's a cutie!!
Sue
(not a Beagle owner but I judged them)