Okay, lets address your points one at a time.
"This war is do or die for Russia, do you think that if they face certain defeat in Ukraine unless they mobilize, that they would fail to do so?"
Russia can't mobilize unless it declares a war. This carries huge risk as it will remove the option of de-escalation. Russia may feel like victory can not be guaranteed by conventional means, however legally declaring war removes the flexibility of an "off-ramp". Russia may want to cash in gains and try again later, without the instability national mobilization would bring. As it stands now though, Ukraine has a head start.
Declaring a war is purely a Russian domestic matter. Every other country in the world considers this to be a war, especially the west.
As far as I understand, declaring war gives Russia the option to draft conscripts. I've seen no evidence that Russia needs to do that, so I agree Russia shouldn't "declare war". Linking it to losing the option of deescalation however is absurd.
"By keeping the tempo of the war at this pace, the Russians can last indefinitely. "
No they can not. Russia's loses are not sustainable over the longer period and we have already seen them dipping into reserve hardware.
Russia hasn't even fully kickstarted their military industry, or called on allies to provide them with military equipment (directly or clandestinely).
The vast reserves of tanks and military equipment shows that Russia can keep this fight going on for years if need be. Russia can get tanks and other platforms to the frontline quicker than it takes NATO to buy tanks from Congo or train Ukrainians to use NATO weaponry.
Equipment is not Russia's problem though, it's manpower. Ukraine has plenty of manpower but lacks equipment.
"While they have a lot of people, there were continuing news that they were grabbing people off the streets and push them to the front with very minimal training and limited hardware."
Ukraine has been doing this since day one.
Which country is going to have a manpower problem, the one with 140 million or the one with 30 million? Not to mention the millions of refugees that have already fled. Ukraine will run out of people to fight far before Russia will.
The idea that Russia is pulling random people off the streets to go and fight in the Ukraine is another western delusion. It is exactly what the Ukraine is doing right now, as evidenced by multiple videos.
Queue for fuel? Get drafted!