My take on parts of your question:
Currently we do have an arXiv, maintained by academia and where researchers regularly upload parts of their work.
I may be on something of a crusade against arXiv users who believe arXiv is more than it is, more more widely adopted than it is. "We" don't have arXiv - certain disciplines have it. Other disciplines, equally valid as those which support arXiv, both don't use it and have understandable issues with the reliance on a pre-print site as a way to disseminate findings.
Why do universities splurge lots of money to publish in third-party journals?
The question especially applies to journals that operate with a rigorous profit motive.
First, they're not paying money to publish in the journals. They're paying money to be able to read said journals. I've published in for-profit journals, even ones my university didn't subscribe to, for free.
The subscription is very high, so wouldn't publishing in such journals affect the paper's citation count and deter the spread of knowledge about the work within academic circles?
Not necessarily. Papers are often available from the author, inter-library loans, etc. Beyond that, how a paper gets cited is a far more complex question than just "Do you have to pay for a subscription", and I don't think Open Access journals have compellingly showed that the citation counts are higher for open journals. The readership and downloads? Probably, but in terms of citation the Open Access journals are still struggling with a perceived gap between their prestige and the prestige of the "leading" for-profit journals. Perhaps that will change in time, but there are ways to get journal articles that your institution doesn't subscribe to, and those ways are often fairly trivial.
Why should not universities collaborate to create free, open access, peer-reviewed journals?
Some do - but for many the cost of laying out and producing a twice monthly journal would be distracting from the core mission of the university (or more likely, particular departments), and they'd run into staffing and budget concerns. Most don't have the money to fund what they actually need to do, let alone add a publishing arm that may or may not ever make money.
And those groups that are interested, like professional societies and the occasional university? They often turn to for-profit publishers to outsource it. For example Epidemiology, a publication of The International Society for Environmental Epidemiology is published by Lippincott. The American Journal of Epidemiology, which is put out by Johns Hopkins and sponsored by the Society for Epidemiologic Research? Published by Oxford.
Moreover, given the need to conserve paper, why should journals spend on printing research papers? Wouldn't an online version suffice, as most people use only local computer printouts anyway?
Because some readers want the paper versions. Seriously, nearly every journal I know has an "online only" subscription for less money. But if you want a paper version, why shouldn't you be able to get it?