Simple answer(s):
In 1941, not all Soviet tanks had radios - usually just the platoon commander. They were more widely introduced as the war progressed, as they realised that tanks without communications were a Bad idea. The Germans had all tanks equipped with radios from the start.
As for comms with arty support: this is more complicated. Most tank radios would operate on one or two frequencies only. The frequencies were set prior to each engagement (a process known as "netting in" in British parlance, for example). A tank commander had a switch allowing him to flick between internal comms within his tank or to the external comms net. So the comms channels didn't get horrendously over-crowded, a single tank company might have one channel to themselves. The command tank might have a second channel available so they could communicate with battalion level.
Artillery assets were most usually allocated at battalion level (or higher). So - as an example, lets assume 4 Platoon, A Company, 2nd Battalion needs fire support...
Platoon commander opens his channel to Company, places the request. Company commander relays the request to Battalion HQ. At HQ, the Battalion commander decides if it is even a good idea. If so, the message is then relayed on to the supporting artillery unit. Artillery unit HQ prioritizes the incoming requests, accepts or denies - and the decision is relayed back down to the Platoon commander.
Given the complexity of the chain of comms in that case, you can understand why the Platoon commander didn't get much of a chance to correct fire...
Now, in certain circumstances, arty assets just might be dropped into the net at a much lower level - for example, at Company level, when the Company had a specific task highly likely to require fire support. In this case, the Company commander just might have direct access to his own personal arty battery - but that assumes plentiful arty assets!
And the other: you will see pictures of specialist arty spotting vehicles. These tend to be de-gunned tanks, or armored cars/half-tracks, with a big stack of radios on board. These could travel with the forward troops, net in with them, while retaining contact with higher artillery head-quarters. These forward observers could obviously direct fire a LOT more efficiently - but they were rare...
Hope that helps some.