I spent 4 months in Cuzco and Urabamba some years ago and I worked in Chinchero. Running in the town of Urabamba would be fine but gets you strange looks as they're not used to really seeing people run for the fun of it. (they wernt then anyway) There are pavements in the town there. Be very wary of running along the big roads out of the centre, they're very hilly - and buses drive up and down the crazily sharp corners at breakneck speeds - often race each other up infact. Theres certainly no pavements out of town and tbh it'd be taking your life in your hands and NOT recommended. Cuzco (and urabamba) is very safe but its a big bustling city and Im not sure how fun it would be running in the streets to be honest. Its hilly, uneven ground and the altitude makes it hard hard work. Le Paz is less safe - I wouldnt advise running there in the dark at all. But thats just my personal opinion.
The way forward I found if you can cope with the altitude (and I cant stress enough how hard that is...!) is to run up from the town centre early morning. Most Peruvian cities have a cross on the hill above the city. Infact almost every Peruvian town or village has this!! Look up and you'll see one. There's a footpath that will wind its way up thats well walked but usually pretty quiet. Beautiful views at the top. Hard core though - steep, altitude... hard going hill run!!!
The other thing I'd recommend is finding the salt flats. For the life of me I cant remember where this was. West of Urabamba - perhaps 10mile away bus ride on the main road towards ollantayambo then a further 10K footpath or so to them each way. Beautiful and a lovely run along those footpaths. Also on the top of the hill towards Chinchero theres a bizzare stone water drainage Incan "pipe" system that runs overground through the most beautiful scenery. Advice is take a map, get off at Chinchero and run east or west and enjoy!!! Theres load of footpaths and back roads everywhere thats much much safer and nicer.
One word of advice - it's COLD at night there. I got caught out one beautiful day up there at a party at the school in chinchero. The sun dropped below the mountains and suddenly it seemed to turn from 20C to -10C in a matter of few hours. By the time we went to leave at 1am there were no busses or taxis. And boy it was cold. Mild hypothermia set in, lips turned blue and out of sheer desperation we ended up knocking on a random house asking for a help in broken Spanish. Old lady of 90 or so took us in and gave us all of her blankets, including the one off her back and led us to a "bed" where we spent the night. Just warning you of my idiotic mistake but am sure you would be much better prepared than I was but get off the hills early. Its a different world at night. (socially very safe but not weather wise!!)
Im really jealous, have a wonderful time!!! Theres loads of Western gap- year students about - you'll soon learn the haunts where they hang out and ask around about runing buddies.