Can I activate Ral Zarek's first ability (Tap target permanent, then untap another permanent) if my opponent has no "untapped" permanents? If so, will I get a loyalty counter, and be able to untap another permanent?
Yes, you can. You must choose a 'target permanent' for the first part of the ability; the ability doesn't specify you must choose an untapped permanent. The ability won't be able to tap that permanent:
701.20a To tap a permanent, turn it sideways from an upright position. Only untapped permanents can be tapped.
but that's not a problem; to activate an ability you must choose valid targets, it doesn't matter if the ability resolves fully. Part of activating an ability is paying its cost, which in this case is adding a loyalty counter:
606.4. The cost to activate a loyalty ability of a permanent is to put on or remove from that permanent a certain number of loyalty counters, as shown by the loyalty symbol in the ability’s cost.
During resolution, even if the first part of the ability doesn't do anything, the second part is still carried out:
608.2c The controller of the spell or ability follows its instructions in the order written.
It's worth mentioning that if Ral Zarek was the only permanent on the field somehow, you could not activate his first ability, and thus not add a loyalty counter to him. This is because the ability requires two different permanents as targets. ("Target permanent" and "another target permanent". Note that Ral Zarek could be one, but not both, of those targets).
This is because (emphasis mine)
602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. ... Activating an ability follows the steps listed below, in order. If, at any point during the activation of an ability, a player is unable to comply with any of those steps, the activation is illegal; the game returns to the moment before that ability started to be activated (see rule 717, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.
602.2b The remainder of the process for activating an ability is identical to the process for casting a spell listed in rules 601.2b–h.
601.2c The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires.