mannii ji
The questions about tea and cold water baths have been answered correctly by Vijaydeep Singh ji.
There is no restriction regarding tea or any other caffeinated beverage like coffee or pepsi.
I am not sure where the question regarding tea came from to begin with. It could be that some individuals or splinter groups generalized from the restriction against intoxicants (narcotic drugs, alcohol) and tobacco.
There is no requirement that Sikhs bathe in cold water. Some sects like 3HO recommend this as an early morning practice, by choice and not by regulation. On this one I did do some research.
Reference:: Sikh Philosophy Network http://www.sikhphilosophy.net/showthread.php?t=7943
The original idea that Sikhs must bathe in cold water goes all the way back to the 18th Century and a document call the Rehitnama by Bhai Nand Lal.
t(h)a(n)ddae paanee jo nehi nhaavai bin jap parrhae prasaadh j khaavai ||
Those who do not take their morning bath in cold water and those who eat before their Nitnem infringe the Rehat.
Rehatnama Bhai Nand Laal Jee
Bhai ji was a close associate of Guru Gobind Singh, and he did write lengthy sets of rules and regulations - many of which are not an issue today. There is no requirement to take a cold bath. None.
Having said this I am myself perplexed why these questions come up. They have been answered. Of course there is no reason why you should take my word for it, or believe vijaydeep singh ji either. So the question that remains unanswered is "Whose word should you take seriously?"
Please read the Sikh Rehat Maryada which covers the code of individual and corporate conduct for Sikhs in the 20th and 21rst Centuries.
http://www.searchsikhism.com/rehat.html
And do be careful not to confuse mention of "bathing" in Sri Guru Granth Sahib Ji to mean a literal bath. Nearly always it is the mind that is to be purified not the body when reference to bathing is made in Guru Granth.
Reference:: Sikh Philosophy Network http://www.sikhphilosophy.net/showthread.php?t=7943 eisaan karehi parabhaath sudhh man gur poojaa bidhh sehith kara(n) ||
After their cleansing bath in the hours before the dawn, they worship the Guru with their minds pure and clear. Ang 1402