I have used cloth pads or cups for ages now and used both at work and at the university. It is really not a problem. For the cloth pads there are bags to buy or make yourself but I just used a system of two plastic bags for the old ones (no smell or leak for all the years I used cloth pads) and a small toiletry case for the fresh ones. For the cup you do not need anything but the cup and the little bag that usually comes with them when you order until you start your period, then the cup is inside you and you just empty it in the toilet when you go. I have used a water bottle or pee to clean it if I have been in a toilet that does not have sinks. My current work place has toilets with a sink and a toilet in a small room so there is really nothing different from the toilet at home so there I rinse it in the sink.
As for cups not fitting, try to research which one is "your" cup there are very different, some are hard and some are soft and they differ in size and length too. I need a soft cup and I have up until recently used the Lady Cup and I am now using the Lunette. The Lady Cup is very soft while the Lunette is soft but a little harder. I find they both work for me. If you cannot find a good cup in the US it is actually worth the effort of trying to order it from abroad as they last for a long time and a good cup is worth a lot. The Lady Cup is from Czech Republic and the Lunette is a Finish product. Within the EU they are easy to order but I don't know how common they are in the US. Here for example the Diva Cup is a bit harder to come by which seems more common in the US.