Question: I work with a group of developers in a company having about 1000 employees. 
A transgender person who identifies herself as a woman has joined our team. 

* She gets upset when we use masculine pronouns (such as "he") to address her, which we obviously do out of habit.
* She also insists on using the ladies toilet, which makes our female coworker uncomfortable.

Taking this to HR is extremely delicate, as they might think we just don't like LGBT people. While we don't want to discriminate against a transgender person, her needs cause issues for us. 

How should we handle this situation?

Answer: 
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> How to deal with this situation?
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People get to self identify, you get to respect that and act professionally and that's it.

People are tied to their identity. It's literally how they define themselves and it's important to them. Imagine it were their religion or political views if that makes things easier for you.

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> such as getting upset whenever we use the pronoun "he"
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I recommend you apologise for referring to a person in a way they found offensive and to correct yourself, as well as make an effort to improve.

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> also demands to use the ladies toilet, which makes our only female coworker quite uncomfortable
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If your female co-worker is not feeling comfortable or safe in her work environment that's something she should bring up to your HR department since that's something that a company should decide and not you.

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As a kind reminder, you don't have to be friends with everyone at your job - if you don't like that person the best thing to do would be to act professionally and also minimize non-professional interaction.
There are a few "simple" things at play here.

Professional behavior, inter personal etiquette / relationships and legal framework depending on the country this takes place in.

There is nothing you personally should do, except to - out of personal courtesy - make an honest effort to use the female pronouns.

The female co-worker may have a chat with her about not being as comfortable to have chit chats in the restroom or to talk about the boys in general.

Depending on her personality, this in fact may be true.

In many countries the employer may be obligated to allow a transgender person to use the toilet of their choosing or decides to allow it to avoid appearing discriminatory.

So escalating it to HR would indeed not make sense and create a more unwelcoming work environment.

On a personal level there is someones behavior towards you and others, their interests and opinions and of course the chemistry as they say.

It is perfectly fine if you end up not socializing with each other.
Don't blow things out of proportion, these things tend to sort themselves out.

Oh, btw. many people make weird noises on the toilet. 
As long as it doesn't sound like someone is in actual pain or seems unwell I see no reason to raise a concern. 

In professional relationships race, gender, sexuality etc. are of no concern and have no place in becoming issues.
What do they bring to the table and how can that support the team or may damage the work are valid professional evaluations of individuals.

In private you may not agree with their personal choices or lifestyle but in free western societies you have no authority to question them.
It might not be your cup o' tea and doesn't have to be, said free society allows you to live your life the way you want it as well...so, yay to freedom and personal liberties I guess (8