To be honest, you'll probably get as many different distros suggested as the number of replies on this thread. I would recommend searching for "best Linux distro" or similar then reading some guides comparing them. That way you can make a decision based on the features that are important to you, rather than the necessarily brief replies you'll get on a forum.
For what it's worth, I use Linux Mint as my only OS and recommend this to friends since this means I can provide local support if needed. If you know any Linux users local to you I'd recommend asking their advice for this reason.
With regard to installing software, essentially all Linux distributions come with a package manager which will, upon you searching for and making a selection of a piece of software you want, automatically download the software, install it and any other programs (so called dependencies) needed for the one you want to work. It will then easily allow keeping that program and its dependencies up to date.
It is nearly always possible (either natively or after adjusting the odd setting) to download a package file and then install it by double clicking, just like in windows. For a Debian based distribution (e.g. Ubuntu, Linux Mint, Debian) this would be with .deb files, for a Red Hat based distribution (e.g. openSUSE, fedora, RHEL) this would be .rpm files. However, it is recommended in the strongest possible terms that you do not do this. There are security concerns associated with installing software from untrusted sources (just like on windows), and by installing the wrong versions of programs you can end up with stuff not working due to also having the wrong dependency versions (so called dependency hell). The only reason to install packages outside the package manager is if the package you want isn't in your distributions repositories (i.e. doesn't show up in the package manager) / some other trusted repository which can be added. That page of instructions you found for Tor describes how to add an additional repository for Tor to a Debian-based installation.