I appreciate good and professional typefaces, but always think that I can improve or customize them for my needs. But where can I design and possibly publish my fonts online?
This is a bit broad and difficult to give a specific answer, but to produce fonts you need software and then to publish fonts start by looking at the largest publishers first and read their documentation.
Independent font publishing is possible via Behance for instance where many freelancers are posting 'home-made' fonts, some of which are quite popular.
Fonts aren't generally designed using online services. I suspect what you are really looking for is font creation software such as fontforge (free), or fontographer (not free).
It's unlikely you will be legally allowed to improve/alter commercial fonts, since these are generally copyright, and as such, making derivatives would breach copyright.
You can publish fonts using services such as dafont.