Ironically, given the perception today, the 'tipping norms' used to be geographically reversed; that is to say, in the United States, tipping was rare, while it was common and expected in Europe. In England at least, the origins come from the 'vail', a practice in the Early Modern period where guests in well-to-do houses would provide a small gratuity to the service staff upon their departure as a thanks for the services rendered (Especially coveted service positions could see the gratuity far outstrip their wages), a practice that lasted for centuries until abolished in the mid-1800s, and eventually found its way into the service industry as a whole there, where it outlasted the ban in houses. Likewise on the continent, tipping was common and expected in the 1800s, although not necessarily the same origins. Interestingly in France, some servers would work for no wages at all in the 19th century, earning only their tips, while in other establishments a 'token' system was used, described by Eeckhout thusly:

Each morning the waiter had to buy a certain number of tokens, which he would then return successively over the day according to the number of drinks sold, but 5 to 10 percent of what he paid was withheld by the owner in order to finance les frais, or general expenses. The waiter's contribution had to be paid regardless of the actual tips he received. A customer who did not tip therefore cost the waiter money.

Anyways, origins aside, it was American travelers to Europe who complained about having to tip back in the 1800s, and in fact, the very emergence of tipping in the American hospitality industry can be found in the switch to the so called "European Plan", from the "American Plan", in hotel accomodations. While in the latter, staying at a hotel meant that food was included in the room price, this was not the case in the former.

When the American hospitality industry was tiny, this hadn't been an issue as far as tipping went. Establishments were almost all quite tiny, and service was provided by the owner and his family. Tipping them would have been seen as an insult. The issue arose in the latter half of the 1800s, as the American hotel industry expanded and it became more likely that service was being done by employees. Under the 'American Plan', owners and managers of hotels did not look kindly upon tips. Rather than being seen as a gratuity for the quality of service provided, they were continually suspicious of guests using tips as a small bride for the service staff, who would in return provide them extra food beyond the regulated amount under their room and board cost.

So slowly, in conjunction with the expansion of the industry as a whole, the transition to the 'European Plan' occured in the American hospitality industry, but to be sure, I do mean slowly. Much debate within the industry over the relative merits of the approaches went round and round, in near endless debates, but the 'European Plan' eventually won. Interestingly, Prohibition was an indirect factor on the final hurdle. Its passage significantly cut into the profits for hotels (alcohol had been extra, of course) and the tightening budgets pushed the remaining holdouts into the 'European Plan' to improve their finances. But regardless, as the shift occurred, tipping no longer was quite the nefarious way the guest was undermining the hotel. Although it wasn't immediately embraced by all hoteliers, as we will touch on below, most eventually came around to the advantages of allowing tips, which could help trim down on payroll and pass it off to the customers to provide! Only in the exclusive high-end joints did the 'American Plan' remain the norm, and tipping firmly dissuaded, at least until the post-WWII era.

Backtracking slightly though, the period from roughly 1900 until 1920, as that shift was occuring, was of course the most contentious. the shift to the 'European Plan' hadn't, afterall, intended to bring tipping into practice, and for management, but especially customers, it was met with reactions ranging from awkward acceptance to outright hostility. In "anti-tipping" hotels, the 'Servidor', a small compartment with a door on the inside and outside of a room, was an innovation to prevent the guests from even coming face to face with service staff. Likewise, rather than have table service, and see guests faced with the dilemma of whether or not to tip, some hotels replaced restaurants with cafeterias. The scourge of the tip was such a pressing matter, that several states went to far as to pass laws banning the practice in that period - although enforcement never was very effective. The laws were short lived, and all eliminated by the mid-1920s, since by then, liked or not, tipping had become firmly established, and the accepted norm. Trade publications, which had been hostile to the practice up until the beginning of the decade, began to at least accept it was here to stay, such as this passage that Mentzer highlights from a 1921 issue of 'Hotel Monthly':

When it gets so that a man pays money to another man in order to retain his own self-respect, which he forfeits if he doesn't, it certainly may be said that tipping has entered the national consciousness"

Again, it was only the very high end establishments that really tried to resist allowing tips after that point. Segrave notes the Grace Dodge Hotel in DC, which opened in 1921 explicitly as a 'no-tip' hotel, and was actually quite successful at attracting clients who wished to avoid gratuities (although of note, it was for women only in its early years), and Longchamps restaurant was founded in 1918 to be a 'tip free' joint, expanding to nine establishments. But these were the exception, and most establishments balked at the higher wages necessary for the approach. And for Longchamps, they more fudged the whole thing, simply having a ten percent service charge on the bill, so more just keeping the patron from having to tip in person, and keeping it formalized on the bill. Some other "no-tip" establishments also tried the service charge route, but it was generally not seen as successful in the US.

So in short, tipping came about slowly in the period from the late 19th century though 1920, in conjunction with the growth of the hotel industry and their changing approach to food service. Now, as for your second question, that deals with much later issues, so I'll leave it to someone else as I must be off shortly (although if I have time this evening I might tackle it).

Eeckhout, Patricia Van den. 2015. Waiters, waitresses, and their tips in western europe before world war I. International Review of Social History 60, (3) (12): 349-378,

Mentzer, Marc S. 2013. The payment of gratuities by customers in the united states: An historical analysis. International Journal of Management 30, (3) (09): 108-120

Tipping: An American Social History of Gratuities by Kerry Segrave

So I'm not sure if you think that tipped workers can be paid less than minimum wage including tips or if you're just stating that a tip is a wage paid by someone other than the employer. But the myth that tipped employees total pay can be less than minimum wage is pervasive enough that I'll at least address that, and it may get you somewhat closer to an answer to your question even if it’s not a complete one.

So let’s just begin with the statement that a tipped employee's total pay cannot be less than minimum wage. The 1996 Small Business Job Protection Act (PL 104-188) set the current regime for tipped workers which is $2.13 per hour plus tips but in any event no less than the current minimum wage. If $2.13 plus tips don't equal that then the employer has to increase the base wage to meet the minimum wage.

But when the original Fair Labor Standards Act (PL 75-718) was passed in 1938 it didn't say anything about tipped workers - though it didn't have to because it didn't really cover tipped workers.

Specifically, the act defined itself as only applying to workers engaged in interstate commerce, and even then there were a host of exceptions, including Sec.13(a)2 which exempted "any employee engaged in any retail or service establishment the greater part of whose selling or servicing is in intrastate commerce."

Taken together, the original Fair Labor Standards Act really only applied to factory workers. The overwhelming majority of the US population, including just about anyone in a traditionally tipped profession, either weren't covered or were exempt.

One way in which the scope of the original Fair Labor Standards Act was enlarged was through the definition of Interstate Commerce. In 1938 the definition of Interstate Commerce was extremely restrictive, but by the 1960's it had come to mean just about every job in the country.

Which lead to PL 89-601, which was passed in 1966. If you look up the legislative history of PL 89-601, which can be found in the debate surrounding HR 13712, you'll see that there is awareness that nearly every job in the country should be covered by the Fair Labor Standards Act but for the exemption for retail and service workers that was essentially preventing the entire service industry from being covered.

So what PL 89-601 did was to remove the service and retail industry exemption. As I stated, prior to this tipped professions were largely exempt from the FLSA and so it didn't specifically address tipping. When they removed the service and retail exception, the way they addressed tipping was to allow an employer to credit 50% of an employees tips to satisfy minimum wage. This would mean that a tipped employee had to be paid 50% of the prevailing minimum wage plus tips, but in any event no less than the prevailing minimum wage. Again if you look in the legislative history, you'll see that Congress was of the opinion that this would be a substantial increase in pay for most tipped workers.

So although it’s not exactly an answer to your question, the minimum wage didn't come into effect uniformly. Originally it only applied to a very small subset of workers, and it was then gradually expanded over the next 30 years to cover the rest of the country. And when it was extended to traditionally tipped professions, those professions largely saw an increase in their total pay as a result of the minimum wage being extended to them.