Update
Salesforce does not allow unmanaged packages to be part of appexchange .You will have to convert them to managed packages .
You can publish your apps as managed package to the appexchange provided you sign up as partner.Do not confuse that partner has to be some consulting firm .Anyone can be a partner Even as a solo developer you can be a part of salesforce partner program .
There are three different types of sign ups
1.Vendor
2.Consultant
3.Marketing Cloud
If you are signing as a Consultant partner there is a some fees .
But since you are trying to build a free apps you can sign as a Vendor and fill in all details .Salesforce will review your form and you can add what apps are you trying to build .Once you are approved you are a vendor partner .You will receive a welcome mail with credentials to log into your p.force.com console .
Here is how publishing console will look
Connect your developer account where you have package to your console
From the package tab you can initiate security review process
Once your unmanaged app passes security review you can create your listing for customers to install
Through p.force.com console,you will get access to publishing console from where you can connect your developer org and create package and distribute package either as unmanaged or managed .
If you are planning to sell the app ,then there are other formalities and you will have to pay for the security review and then contact salesforce for various revenue sharing options.Even to list a simple unmanaged app your app will go through strict security review process but Security review will be free of cost .
Best Answer
There is no necessity that if we are making a managed package then it should be on Appexhange. Appexchange is basically a marketplace for salesforce as Itunes is for Apple.
To install a package:
When create a package and upload it, after successful upload it provide installation link. It can be gone through Build | create | package and select your packages > Then select your package name > Installation URL seems like https://login.salesforce.com/packaging/installPackage.apexp?p0=04t900000003mGs.
Just copy and paste that url and follow the instruction to upload.
Note: For successful package creation test class coverage must be >= 75%