Keeping anything critically important in only a free email account like Hotmail is unfortunately a recipe for disaster.
I want to go over how you might have been better prepared for this. I’ll touch on what you might do to get your email back (but to be honest, I’m not very hopeful) and I’ll talk about the service that you let access your machine.
Putting all your eggs in the Hotmail basket
Over the years, I’ve seen many important documents and emails get lost in free email services. Sometimes it was the service’s fault, and sometimes it was the user’s fault, but they all were lost – forever. Free email services have little if any customer support or recourse to help you out.
The key here is that the stuff that you consider important was stored in one and only one place: a free email account. The way you could have prepared for this is actually very simple: back up.
There are several ways to back up an online email account, and had you done any one of them, this would not be the disaster it’s very likely to turn out to be. So, if you walk away with only one lesson from this, please let it be back up.
Getting your email back is a little bit harder. First I’m going to point you at an article I have called, “How do I contact customer service for Outlook.com?” There are several options covered in the article.
Ultimately, there are no people to talk to and no emails to send, but there is a website that you can go to that has a lot of frequent scenarios. There is an item there called “restore deleted email messages”. All that really is is just looking in the deleted folder to make sure that they’re not there. I do, of course, recommend you do that.
At the bottom of that page they also give you a link to a contact form. It’s possible that someone will get back to you with assistance on the issue but like I said, I’m not very hopeful. Another option is to ask for help in the support forums also linked to from the customer service article I mentioned above.
Remote access support
About that company you contacted: it does sound like you mistook an advertisement somewhere for something more legitimate like Microsoft.
I’m not familiar with the company you mentioned, so I can’t say that they did any damage.
I would definitely keep an eye on your machine for suspicious activities, and I would make sure that your anti-malware tools are running and up to date. I would also make sure that your computer has up-to-date databases of malware information.