I am so sorry you had these outcomes. MTX can be a significant help for many but it can also be a difficult medication to use for some.
I have used MTX for over 20 years and have experienced some, but not all of the issues you describe. I have had the awful mouth sores, and I have felt like I have the flu after the weekly dose. I have not ha blood issues but I have heard of that complication.
Over time I changed my method of introduction form the pill form to injections. I also started using Leucovorin in addition to the MTX.
Leucovorin is a highly concentrated folic acid, available only via prescription and that is usually used by cancer patients. From the time I started using Leucovorin I have not had additional issues.
Two notes, there are other immuno suppressants that may work better than MTX for you. Ask your doctor if they might suggest an alternative to MTX. In addition if one has mouth sores, you might consider asking your dentist or rheumatologist if a compounded medication Mary's Magic Mouth Wash might help out. I found it very effective for mouth sores before I stated using Leucovorin.
With regard ot the flu symptoms. If you did have the flu MTX may not be appropriate to use. Again ask your rheumatologist. MTX is an immuno suppressant, meaning it may be more difficult to shake the bug, depending on the type. I suggest you ask your rheumatologist to work out an protocol plan from when you have a bad cold or flu.
I hope you get to feeling better soon.