The main problem with the rom category, is that it serves to describe as one blob a multitude of different societies and cultures, and you illustrate very well the confusion that it allows. You claim to designate only a specific romanian group (romani ? that is the collective language, spoken in majority, that is supposed to link these groups), and yet you jump to a german lutherian citation about gypsies, with zero context, simply postulating identity. First of all, the politics of mendicity and mobility in the middle ages were in a huge crisis as the legitimacy of charity was being strongly redefined through geographic boundaries. There was an explosion of edicts and discourses narrowing the "okay to give to" versus "okay to slaughter" categories of beggars, and non-locals were designated as the big no-no, unholy on all levels. This meant people coming from other regions, and of course, all nomads. Should a beggar be a gypsy, of course he'd be on the "bad beggars" list. He wouldn't be alone. But that's not even the main point. The point is that you cannot hop from middle ages gypsies to romanian rroms and infer cultural identity (the ongoing tradition of mendicity), because not only you hop from one geographical era to another (from german gypsies to your i-only-talk-about-local-romani), but you also hop above centuries (no logical inference of european traditions of witch burning ?).
This shortcut jump is possible because you use "rom" as some intemporal homogeneous abstract entity. Negating all it means, in favor of one stereotype which is far from reality : the nomad beggar. And this is what skips the fact that 95% of rroms are sedentary. And it skips the fact that most of them rely on other means of subsistance than mendicity. ecause most of the people who belong to the 'rrom' category do not correspond to the very stereotype that the word evokes to the general public.
As for the caste thing, I assume that you looked for it the same way you looked for the Luther quote - a quick google to a sentence with no context (a trocated footnote down the googlebooks page of a book you never opened), and possibly stumbled upon the touristic tzigana site that shoehorns indian castes into rrom socioculture (diversity doesn't mean castes) for the very same reason you attached that middle age quote : random cultural inferences sound super awesome.
But what matters, and what you should rather turn your attention to, are contemporary ethnographies of 'rrom' groups. That is, accounts of actual long-term participant observations, describing from within the actual "contents" of rrom everyday life, and the actual life trajectories of real individuals and families. Or else, you stay at the exact same levels as contemporary greeks who only describe themselves, their country and their societies, through references to ancient greece and byzatium (studying ancient greece will tell you absolutely nothing of the current greek society, no matter how many romantic cultural inferences one would like to draw). Either that, or you stay at the outside level, projecting various stereotypes and popular discourses on whatever glimpse of 'romani' society you see (dirtectly or through news).
As for child beggars, no, they are very rare in Switzerland and possibly west Europe, even though they are more frequent to the East (in Greece for instance), which doesn't meant they are victims of exploitation (nearby gypsies camps, squads of kids enthousiastically play "get coins from the tourists", and in many other contexts they are simply accompanying their parents - though you can raise the issue of obligatory schooling and its costs/benefits). The only kid I've seen in Switzerland was accompanying his brother, pretending to be mute in a typical (but minoritary) NGO fund-raising scam.
You can relate mendicity as one traditional venue for certain subgroups (for instance the largely minoritary nomadic ones). It is also a survival strategy for a certain category of jobless people from post-communist east europa. But attaching mendicity as a core cultural feature of the abusively blurry "rom" category does not describe anything else than an enduring, reductive and essentialist stereotype. Even in Romania, where the agricultural Gabori used to work as blacksmiths, tinkers, day labourers, workers on building sites, etc.
And there are actually two issues with that. First, in political discourses, mendicity gets often assimilated to rrom identity, regardless of local beggars. This merges the "illegitimately foreign" stigma with the "pauper" stigma, and to further stigmatize poverty. The other issue is that, even without that, mendicity itself is declared illegitimate and immoral. That is an underlying conception that further fuels anti-rom sentiment (dirty beggars, all of them), in addition to make things even harder for fragilized people. This has its sources on other cultural elements (the life-earn-through-"real"-work ethics, the scandal of visible misery as better hidden symptom of society's failure, the breaching of the sacrosanct street indifference that should allow us to ignore each others outside, etc), but it is still an arbitrary judgement that complicates the stakes of rrom-related discussions and policies. And it has been largely illustrated in this thread.
Maybe a certain distanciation from this judgement on mendicity would be as important as an actual assessment of "what it means to be rrom" (in practice, and for whom) nowadays. Because both issues seem to strongly enter in resonnance...