The Imperium taxes worlds based on the specifics outlined at the time of its induction into Imperial membership. It is the responsibility of the world government to see that the taxes are collected and paid in whatever means it chooses - that is a matter for local affairs, and the Imperial Noble overseeing the world sees to it (via oversight of local and/or Regional Branches of Imperial Ministries) that they are in order and channeled into the Treasury via the Finance Ministry.
Taxation of interworld trade occurs as well, of course, but that is interworld commerce (also overseen by Imperial Nobility via Imperial Ministries).
The Imperium "ruling the space between the worlds" is a euphemism - the Imperium is not (and never has been) described as a Confederation. The phrase simply means that it doesn't rule the worlds directly and can't be bothered to deal with governing at such a minor level - it leaves that detail to local concerns. But those concerns most certainly are in subjection to Imperial interests and authority; otherwise, the Imperium does not rule at all save by mutual consent - and that is not a superintending government, let alone and Empire.
The Imperium does not interfere in local interests, government or culture as long as they do not conflict with Imperial interests. When they do, the policies pursued by the Imperium to correct the issue may vary depending upon the local situation after "discussion" with local Imperial Nobility and other relevant officials - the Imperium has at times been more subtle and patient (i.e. devious) in response, or "Kerrian" as Cleon Zhunastu and Zuan Kerr originally envisioned, and at other times more "Artemsian" like the Lentuli Dynasty and it's iron fist.