I'll address your specific examples.
Player B says he is tapping his Steel Overseer to put a +1+1 counter on all of his artifact creatures. Can I tap it first to prevent him from doing so?
If the tap is part of the cost of activating the ability (ie. tap symbol is before the colon) then the answer is no. You can't stop a cost from being paid. If you put an ability on the stack to tap it, they could tap in response. If they tap it to activate the ability, the cost is already paid, and tapping it in response does nothing.
In your preamble you seemed to indicate that in the past you could prevent activated abilities with a tap cost by tapping them first. The truth is, the game has always worked this way, even before the addition of the stack. Back in the day, the last person to cast contradictory instants got to determine order of resolution. There was even a ruling, back in the mists of time, clarifying that if you tapped something that had already been tapped, it has no effect on the action.
You could never twiddle a Prodigal Sorcerer to prevent it from dealing 1 damage to your dome.
Player B says he is tapping his forest to add a green mana to his mana pool, can I tap it first to prevent him from doing so?
You cannot, nor could you ever.
If I cannot, then what use is my ability to tap his permanents other than to prevent him from attacking or to take advantage of his tapped status (either by counter-attacking or perhaps an 'assassination' effect)?
What use? Tapping stuff, even if it doesn't prevent activated or mana abilities is still very very useful. Preventing a creature from attacking is usually quite important. It is also extremely useful to tap a defender after you declare attackers to give them a significant advantage. Especially if said defender has, say, deathtouch, first strike or regeneration.
This isn't to say that you are necessarily incorrect in your assessment of the utility of the tap effects in your particular deck. If your deck already keeps the ground clear by a constant barrage of direct damage or other removal, or your big beaters have trample and don't particularly care what gets in the way, by all means, remove the tap effects from your deck.
Also, if I cannot, is there a point in a turn when I COULD tap his lands and accomplish starving him of mana? If in theory, he could always tap his lands for mana in response to my attempt to tap them, how long would such mana remain unused in his pool until it expires and he cannot use it?
Comprehensive rule 106.4:
106.4. When an effect produces mana, that mana goes into a player's mana pool. From there, it can be used to pay costs immediately, or it can stay in the player's mana pool. Each player's mana pool empties at the end of each step and phase.
So you could tap the mana during that player's upkeep, and that will deprive them of using that mana during their main phase for sorceries or summoning a creature. Or you could tap their mana during their end step and that might prevent them from using nasty combat tricks or counterspells during your turn.