Don't listen to your instructor he is not a doctor, but do get the class 1 before you start. As for max height it is not 5' 11 more around 6' 4, I am 6' 2 and had no trouble in getting a job.
Sponsorship you can pretty much forget about for the next year to 2 years, there are far to many experienced type rated guys out there who are looking for work and until the market improves which is gonna take some time the airlines will look to recruit the guys with experience first and until that pool of pilots dry up you may be better off paying for it yourself otherwise you may end up waiting a long time