Iis – Problems with Exchange services (AutoDiscover, ECP, OWA)

I have some problems with Exchange 2013, but first you will need to know how this happened:

Background

About two months ago, I setup an Exchange server (2013 SP1 on Server 2012 R2 box, virtualized on ESXi 5.0). Everything was working fine till some days ago, at time which I decided to install the Web Remote Management role on this server (Yes! I know I shouldn't have done this, but I didn't know anything about its side-effects). After enabling that role, the entire Exchange service went down.

I resolved it by removing that role, but only Outlook connectivity was restored. ECP, OWA, PowerShell and any other services based on IIS didn't came back up. After many hours of work on IIS, I could bring back ECP and PowerShell up. My work included resetting all virtual directories (on Default Website) and playing around with certificates.

Current problems

Now, the problems I have:

1. Yesterday, suddenly some clients lost their Outlook connectivity, reporting that it says: "Trying to connect …" and after some time: "Disconnected". This problem propagated to the entire company yesterday and today and all clients faced this problem. The only solution we found to this was to remove/repair accounts, including manual setup of accounts. Outlook AutoDiscover doesn't work at all. At 3rd step on setup, it says The connection to the Microsoft Exchange Server is unavailable. and asks for server and mailbox address. We have to setup Exchange account manually and enter IP address of server and mailbox address (It even doesn't work with FQDN, though DNS lookup on the same client and at the same time results in successful resolution of FQDN to IP). The strange part is that after setting up account, if I right-click Outlook account in system tray while holding CTRL key, and then choose "Test E-mail AutoConfiguration…", everything works fine there and tests state that AutoDiscover works!
2. In random times, Outlook asks clients for their Windows credentials. I don't know why, as before this it never did so.
3. When entering ECP, I face HTTP Error 404 (IIS error page), titled Server Error in '/ecp' Application. saying that Requested URL: /ecp/login.aspx not found. But when I clear everything after /ecp/ from the URL appearing in the address bar (https://exchange.golbarg.local/ecp/login.aspx?ReturnUrl=%2fecp%2f) and hit ENTER, ECP opens successfuly and works fine.
4. A problem similar to 3 appears when entering OWA. A page saying 404 can't find page :-( appears (Outlook error page which is different from IIS one).
   Again, clearing everything after /owa/ from address (https://exchange.golbarg.local/owa/auth/errorFE.aspx?httpCode=404#ReturnUrl=/owa/?bO=1&bO=1) gives OWA working fine.

Any suggestions are welcome.