1) Is there a genetic basis for these disorders? Do they run in families?

Yes.

2) Is that genetic basis MTHFR?

No.

3) But I "have" MTHFR, and a lot of other +'s and really think these mean something important.

We all want research on rare diseases, genetics in general, neurochemistry, and a lot of other fields to be at Star Wars level, and instead, these fields still feel very much (especially if you fall into some particular subpopulations who need more answers, now) like they’re at Flinstones level. Many of your “plusses” might be shared with other people with these conditions...but it’s very unlikely any of them will change your treatment plan in any respect, with the exception of some forms of B vitamins you might be taking in the case of MTHFR mutations and some others you should avoid (probably either way). You’ll meet a lot of very excited people in support groups who come for advice because they’ve just found some of these. Most commonly, they’ve discovered they “have MTHFR.” In many of these cases, they’re in good company; they have the mutations that at least 40% of the population have and are going down the wrong research rabbit hole. Also, just like many other genes, “MTHFR” is a lot more complicated than B vitamin forms. You could be an over-methylator or an under-methylator, and well-intentioned attempts to “treat” the thing you found in your genetics results is, by my observation, about equally likely to cause an overcorrection or to go in the wrong direction to begin with and leave you in a much worse situation. If you have concerns about your profile, please seek the help of a qualified geneticist. Most of them won't feed you to their pet pterodactyl.

4) Can I safely get pregnant/harvest an egg/carry a child to term, or will this make a mast cell disorder/related conditions worse? What are the chances that my child will have a case like mine?

Regarding pregnancy/eggs/carrying a child: I of course can't answer this for your own case and strongly recommend you discuss this with a mast cell specialist (a specialist, not a familiar provider) and a high risk obstetrician who deals with several patients in this population. In general, though, you should know that hormonal fluctuations, especially quick or dramatic ones, periods of extreme physiological stress, and surgeries are among the most common "big" triggers that incite initial or worsening mast cell symptoms. Some patients find pregnancy unsuccessful or debilitating. Others carry children safely, or even feel more stable during this time than they usually do, if other hormones are their triggers, but their condition may worsen significantly afterwards (another time of dramatic hormonal changes and, in case you aren't yet aware, just procuring a baby human tops the list of stressful life events.) With POTS or other dysautonomia, vasovagal issues are more common during pregnancy for many patients. With EDS, hormonal changes themselves may also be challenging, but more worrisome tends to be the effects of relaxin, a hormone secreted later in pregnancy that relaxes connective tissue throughout the body, as with faulty collagen production, EDS patients are less able to repair or reverse these changes and maintain connective tissue integrity throughout their bodies. For this reason, generally, EDS and other connective tissue patients, assuming heart (aortic) conditions are not of more immediate concern with a pregnancy, are known to be at greater risk of cervical incontinence and especially later term miscarriages, uterine rupture (a risk of fatality to the mother, as well as to the child), prolapsed organs, and general worsening condition.

A little more on the genetic basis: Certain variations in one gene have been identified in most mastocytosis patients (others are also very likely), and MCAS and trifecta conditions in general strongly run in families (EDS, HAE, HATS, and certain forms of dysautonomia are all genetic disorders, though the genetic test for the most common type of EDS is not yet widely available). It's rare to find a symptomatic patient whose family history is not indicative of the presence of one or more of these disorders, though it is common to find that earlier generations presented with less serious symptoms than do later ones. There could be a number of environmental and other reasons to account for this, and likely are, but I'm not the right person to speculate about them all.