Yes. Any lump should be examined. It may be something as simple as a corn, wart. Mucoid cyst etc. But, a lump is not normal and since we can't see it - have it checked out. It will give you peace of mind. A podiatrist would be a good doc to see for this.
You. May have a hammer toe. See a podiatrist I am sure they can help you.
Yes. This is most likely a corn or callus caused by wearing uncomfortable shoes. It could also represent something worrisome. Have a healthcare provider take a look so you know exactly how to treat it!