I've stayed in hostels in Europe around 93 nights (quick math in my head) and in the U.S. in New York, New Orleans, Chicago and Denver around 15 nights total. I'm 76 and have been solo on all these trips, which have ranged over the last 7 years.
Did I feel safe as a solo traveler? Yes, quite possibly safer than I would have felt in a hotel, as there is more of a community feeling.
Did luggage feel safe? Yes. I carry all of my true valuables (money, cash, passport, driver's license) on my person in a money belt at all times. There is always somewhere in the room to lock up things, so often I lock up my tablet that I bring along. The rest of the items feel safe--no one wants your dirty clothes. Now if you are a fashionista and have very expensive clothes or shoes that you intend to leave in the room, or a very expensive camera, that might be different.
Would decision depend on the country? No.
Does it save over a "standard hotel"? Yes, in 100% of the cases where I have stayed thus far. I had to forego the Greece portion of my trip in early April this year; that was the only planned stay where a hotel might have been almost as little as a hostel. Consider that often you are paying 20% or 25% of what you would pay for a hotel. In over 100 nights, there has never been an instance when I had to pay for sheets, although I believe it is still possible. Any fees like this would be disclosed in advance of booking. You do not have to pay to rent towels either, if you invest around $10 in a "camping towel." The one I took this year dried in literally about 20 minutes after showering, so I did not rent any towels. However, if you can't stand that, it is nearly universal to have to pay a small towel rental fee. There are no guarantees about bedbugs, even in a hotel, but if you read the reviews carefully, you would most likely find out about those infestations. If a hostel doesn't have recent reviews, I don't book it.
Click on my name to read my reviews of two long trips, where I describe the different hostels I used. I have not found Hostelling International's hostels to be any better than the unaffiliated ones. I strongly suggest using the hostelworld.com reviews. If you are middle-aged, senior, or a quiet young person, these generally will tell you if the hostel is "party" or "social," and you may want to avoid those. One big advantage is the ability to interact with people from all over the world. Some hostels plan interesting excursions that can save you from having to pay for a private tour too, they tend to be located very near train and/or transit stations as well as lower-cost places to eat, and some offer breakfast and/or dinner at lower than market rate costs.
Can they be noisy? Yes. The same as a fancy hotel can be noisy. Can the bathrooms be a little dirty? Yes, although they are disorderly much more often than dirty as such--in fact I do not remember one that was dirty but if a little water on the floor and some people not hanging up the bath mat just the way you like are going to ruin your trip, you may not be hostel material. Read reviews, take the complaining ones with a grain of salt just as you would negative hotel reviews, and you should be fine.