Yeah, no solid evidence to support the Templars morphing into the Freemasons through some type of hokey lineage nonsense. The Templars were monks, sworn to celibacy (seriously, they weren't even allowed to touch, or be in the same area alone with a woman). Now anyone who knows the first thing about biology will be able to inform you the problems with claiming a bloodline to a bunch of celibate dudes.
Templar basically joined the Hospitallers, or became Knights of Christ, which was literally the Templar Order with a different name. Nations were usually pretty ok with this kind of thing going on, since they were only found guilty in France, and were found no guilty after they were already disbanded and burned. Awkward.
Ok, short history of the Teutonic Knights.
They were founded during the 3rd crusade at the siege of Acre. At first they were a branch of the Hostipallers, but soon because their own order, meant to blend the Templar's military ferocity and organization with the Hostpitaller's charity, with a unique German flair. They, while maintaing castles and supporting campaigns in Outremer, organized and led the Baltic Crusades against Pagans who were ravaging Poland at the time. They were wildly successful, conquering and holding their own nation state, and becoming powerful enough to rival any European nation. They were undoubtably the most powerful military order, with their own army (which was rather large, even among nations, for the time), diplomats, traders, and chapterhouses throughout Europe in addition to their sovereign state.
At the zenith of their power, they succeeded in their Crusade efforts, and converted the last of the Lithuanian Archdukes in 1389. Shortly after though, they faced a decline as nations (specifically Poland and Lithuania) grew stronger, and united against the Teutonic Order. This, combined with the growing anti-monastic sentiment throughout Europe left the Order without men, and with many enemies. They were eventually driven out of the Baltic in the 16th century, the Livonian Branch being wiped out by Ivan the Terrible in 1545. They were only officially suppressed by Adolf Hitler. After the fall of the 3rd Reich, they were reformed as a religious/charity order, which they function as to this day.