What I've done for myself when I had phone based internet service...including going back to the dial up days...and I have my clients do that same thing.....when getting DSL...have the phone company run in brand new lines from the street...to the house/apartment/building...all new lines right up to where the DSL modem will be located. It really won't cost much..and the greatly increased reliability is well worth it. Did that even back in the dial up days. During bad weather I'd have no problems and I listen to others moan and groan about connection problems in bad weather.
Other alternative..have cable in your area? Usually quite a bit faster than DSL..sometimes a little bit more expensive. Based on your needs I don't think you'll find cellular internet very good for you.
Regarding fax needs..plenty of internet based Fax setups like eFax...works with your e-mail client...my wife uses that for her home business.