1) You are right. There are many combinations of よろしくおねがいします AND/OR はじめまして. I think the standard template of introducing is:
If you just put your name instead of 田中 in your response, it will not be bad. But you can change the template depending on the situation. If you meet a person who has social status or age same as you (friend of your friend, for example) you can skip はじめまして or よろしくおねがいします. Also you can say simply よろしく or どうぞよろしく(hardly used) or よろしくね instead of よろしくおねがいします. It sounds friendly but be careful and don't use it when you meet elder person.
You can use also といいます instead of です (only in case of introducing)
[私は]田中です – I am Tanaka
私の名前は田中です – My name is Tanaka (hardly used but can be useful depending on situation)
[私は]田中といいます – I am called Tanaka (very verbatim but japanese often use it)
Note that 私は(わたしは) is skipped usually – no sense to say "I" when you are talking about yourself (in japanese thinking)
Meeting elder person. If you want to be very polite in introducing, you can use next variants
でございます instead of です (hardly used but sounds very politely)
ともうします instead of といいます
よろしくおねがいいたします instead of よろしくおねがいします
Remember: you should use polite words throughout the sentence, not only one.
2) You said "The general response to はじめまして is よろしくおねがいします". But it is not quite so. My advice is - dont think rigidly. Use it appropriately to situation.
What does mean はじめまして and よろしくおねがいします?
はじめまして inherits from verb はじめる(begin, start) and it is just greeting to person, who you meet first time. It probably means something like "let's begin a friendship" (very verbatim again
(source: kolobok.us) ). よろしくおねがいします does not mean "nice to meet you", "i'm glad to meet you" or somethink like this. おねがいします means request to do somethink like "please do it for me" and よろしくおねがいします is just more polite request. I don't know what japanese are requesting to person in first meeting (probably it is like "please be a friend for me" or "I hope you will be kind to me". It's not more than idiomatic expression) but it is often used in daily life. Example:
A: Can you do it for me?
B: Yes, I can
A: Well, おねがいします
(english-japanese mix )
3) You said "The general response to よろしくおねがいします is こちらこそ". Yes, it sounds good but needs some addition. こちらこそ sounds simply, so you had better respond with full sentence こちらこそよろしくおねがいします if you want to be polite. Also you can put いいえ before it (いいえ、こちらこそ[よろしくおねがいします]). Why いいえ? Try to remember The Matrix scene where Neo meets Morpheus first time:
Neo: It's an honor to meet you
Morpheus: No, the honor is mine
(Note, that is not translation, only analogy).
こちらこそ is just emphasis to yourself and can be used in many situations. For example, when some person helped you, and you helped that person, and when he/she says "I'm grateful to you", you can respond "no, I am very grateful to you" with using こちらこそ.
Here are a direct answers to your questions
1) "Can I respond to はじめまして with こちらこそ?" No
(source: kolobok.us) respond to はじめまして with はじめまして and add after that こちらこそ if companion said よろしく. Don't use it when you talk first and don't say it if companion did not say よろしく.
2) "Can I respond to よろしくおねがいします with よろしくおねがいします?" Yes, you can, but こちらこそよろしくおねがいします sounds a little bit better.
3) "*If someone says 田中です。はじめまして。よろしくおねがいします、 can I respond with something like こちらこそです?" It has been already answered above.
And finally I want to add some wrong examples
Companion: mmm... What こちらこそ
Companion: mmm... What こちらこそ
So, as you can see, japanese is not rigid but very flexible (sometimes too flexible to understand or remember all these
(source: kolobok.us) )
Good luck with your study. がんばって