Aspirin cause intestinal bleeding. Period. No question about it. Used to kill many thousands of people a year. But that doesn't mean you should never use it, it just means it should be used in moderation, only when needed. Other painkillers are also toxic, particularly liver toxic, and can also cause internal bleeding. Ulcers, while associated with h-pylori, have a cause and effect problem. Since h-pylori is found in virtually everyone without ill effect, the question is why it manages to find good lodgings in some and not in most. Aspirin is one contributing factor. Blood thinning pharmaceuticals do have their own problems as well -- look at all the contraindications with all of them. There are no safe pharmaceuticals, it's a cost-benefit analysis of whether it's the only thing you can do about the problem since all synthetic drugs will cause the body to operate in a way it normally wouldn't, and have to force themselves past the body's immune system to work. While aspirin is derived from natural sources (white willow bark), it is pharmaceutically isolated and much stronger than in any natural source and has always been a source of thousands of deaths a year since records of such things have been kept. Which again doesn't mean it still isn't the best available choice in a given circumstance, it just means caution is necessary, and milder natural remedies can be tried first if the problem isn't acute.
Ulcers are commonly caused by a bacterial infection - H.pylori, that exists and thrives in high concentrations of acid.