Hello everyone

This is my first thread on this forum, I'm coming from the iPhone 3g, and just bought a new HTC Desire S... So I'm a total newbie..

I am very happy with this phone, still, I have few things that I don't quite get,
and I would be very grateful if you help me with them..
I have problems with: setting up email / not getting notifications for mail in realtime / not getting notifications for facebook messenger in realtime / getting annoying google talk authentication notifications
- Firstly, I'm having a problem with setting up the e-mails.. I have 5 emails that I use, and 4 of them are on gmail. 3 of my gmail accounts are with @mycompany.com domain.
So, I have:
mail@gmail.com
mail@mycompany.com (gmail)
mail@mycompany.com (gmail)
mail@mycompany.com (gmail)
mail@yahoo.com
1. I first tried to make accounts on the basic "mail" app, but I noticed that my e-mails are not syncing with my account at all. The messages are not downloaded, and I am not receiving mails neither on the regular gmail, neither on the mycompany gmail. When setting up the regular gmail (new account, automatic) I also got the message: "Warning: Cannot connect to the remote server. Please check the network settings and then try again". Why?
2. I then used the gmail app for emails, and successfully configured all my 4 gmail accounts. One problem is that I don't understand how to check them in one click (like I used to do on my iPhone on the refresh button). Do I have to manually click on "Sync All" button? Even if I do that, I've noticed that that is syncing all my accounts (including facebook and twitter, and contacts.. and everything.. and it's taking a while..)
3. The other most importaing thing to me is - I don't understand why I am not getting notifications for e-mail in
realtime , or when I just touch the lock button, I used to have that option on my iPhone, but why I can't have that here?
Message actions are on..
e-mail notifications are on..
So why I am not getting e-mails in realtime?
PS. I use whatsapp, and I'm getting messages from that app in realtime..
4. Next important question is: Why can't all my gmail accounts be in one inbox?
5. I am getting this message all the time: Google talk authentication failed. I don't care why, I just want it to stop! Does this have something to do with my @companymail.com google account?
6. Notifications for the Facebook Messenger app. Also, not receiving them in realtime. Why?
Thank you for your time