The arr 1-20 fan out clockwise but in snail shape.
I think most of Paris proper is safe and relatively close to everything.
I do prefer the 6th arr because of the nightlife; tons of restaurants, shops and bars open till late.
The Latin Quarter is a bit touristy and noisy but does have a few nice places to hang out and even a few nice places for an inexpensive simple meal.
As far as hotels, usually you get what you pay for.
Prices have increased significantly in the last 20 years even though the property may have NOT been remolded in the last 20 years.
PM me if you want any more specific suggestions.