Yes, cloning, repeatedly, maintained, could make you immortal.
There's some logistical concerns that make this trickier than the spell itself:
Vessel must be undisturbed
So, ideally, you set up a nice young version of yourself, hide it away for the time something goes wrong and go about your life, right? Well, the longer it's around, the more likely, over time, something COULD happen to it. Especially in a world where you've got things like purple worms, umber hulks and bulettes and other critters that dig through granite like butter.
Well, then it makes sense to set up some defensive measures, right? Traps, spells, etc. But in the world of D&D, the more defensive measures you put up, the more people assume it's got something valuable to steal...
Now, as a GM I wouldn't just automatically assume something is going to happen, but if the clone is sitting around for decades, or the wizard in question has enemies seeking them out, then we'd start having to think about problems.
A giant diamond
So, the diamond is worth 1,000 gp. This doesn't mean you can simply pull out 1,000 gp and find these diamonds anywhere, everywhere. There's got to be a limited number of them. You're probably not the only caster who is looking into this spell.
So, a bunch of wizards want immortality from a limited resource and are all looking for it.
I'm sure that's not going to lead to problems.
Welcome to the Immortal Club
So, if you manage to live far beyond even what most D&D world folks know people to be capable of, and you're known to be an awesome wizard... how many other people are going to be trying to get your secret of immortality from you?
How do other things which are immortal feel about this? Do they find a way to manipulate/play you because you're new to this game? Do they already have a control on the 1,000 gp diamonds and dole them out to the few wizards who have Clone just to keep them under their leash?
Is there an alliance of lichs who are jealous you've found a way to live, but actually live, not undead live, and they'd like to simply stomp you down for being audacious?
Are there mind flayers looking to eat the juicy mind of a super-intelligent wizard with 800 years of tasty-tasty knowledge?
Are there divine guardians of life and death who did their accounting and finding there's a soul short that needs to move on?
Gameplay
Unless you're playing a very unusual game of D&D, these issues aren't likely to come up too much simply because the timescale is too short. But it makes excellent source of adventures based on NPCs - just imagine what happens when you do have a wizard who has been doing this and dealing with all of these problems and what that means for the PCs when they get involved in it.
As far as I know, racial spells aren't actually defined anywhere, and so the following answer is entirely my opinion, well-reasoned though it may be:
For your examples, the answers are yes, yes, yes, and no.
Racial spells are only considered "default" or commonplace for characters of those races. This means that a human wizard can't select aboleth's lung as a "free" spell when leveling up, and that a human sorcerer could only select it if they had access to the spell somehow (such as studying with a spellcaster who had it). Divine casters would need to be at least made aware the spell exists before they request it from their deity. Basically any other means of learning a spell besides just leveling up should remain valid:
- Learning the spell from a scroll, spellbook, or other spellcaster
- Researching the spell independently
- Identifying the spell as it's cast with a Spellcraft check (for casters who only need to be exposed to the spell, like sorcerers and most divine casters - this isn't comprehensive enough for a wizard to add the spell to their book0
- Suffering the effects of the spell (for caster who only need to be exposed to it)
- Learning about the spell from a magical source, such as speak with dead or via an outsider
All of these things constitute "GM permission". If the GM doesn't want you learning the spell, all he has to do is not introduce a gillman wizard with aboleth's lung in his spellbook. Not already knowing the spell exists, you couldn't really call up a water elemental or devil to ask about it directly, and the GM is in control of spell research as well. Buying a scroll of the spell is impossible if no one knows it to scribe them in the first place.
If the GM doesn't mind you learning it, then it's safe to assume that the spell has made it out of its circles and into a more general curriculum. It could be anywhere from as hard as actually going to a gillman village and buying a scroll (for an inflated price, even) or as easy as simply choosing the spell on level-up, depending on your GM.
Best Answer
Make it a Downtime Activity
More on less on par with Crafting a Magic Item.
This is how I approach permanency in the Eberron game that I run, and my players have been happy with it so far.
The initial problem
Trying to directly port Permanency into 5E is a problem, since Permanency had an XP cost, and using XP is an optional rule in 5E.
Precedent
D&D 5e's Adventurer's League has a precedent for charging Downtime as a 'toll on the character's soul.' (See Here, the section on Jenny Greenteeth)
For further precedent...making a magical effect permanent is essentially what you are doing while crafting a magic item. So making a spell Permanent is following the same sort of procedure as you'd use to craft an enchanted item with a persistent effect...but you're attaching it to a person or place instead.
My Rules
For the downtime activity of making a spell Permanent, I basically follow the rules for crafting a magic item, starting on page 128 of the DMG, using the 'Power By Rarity' table on page 285 to determine the 'effective rarity' of the spell to be made permanent.
Because Permanent spells can be torn down with Dispel Magic, I treat them as 'consumable' items (half the cost and creation time of a 'normal' magic item of that power level). If you want Permanency to be harder...then treat them as normal cost for their rarity level. And if you want to cap which spells can be made permanent, bear in mind that 3.5e actually had a rather short list of spells that could be made Permanent.
By dint of flavor, I generally say that making a spell Permanent doesn't actually take the full duration of you working on it. I rule that some of that time (often about half) has to be spent recovering from the physical and spiritual exertion of what you just did.
Requirements
Naturally, you have to know the spell in order to make it permanent, and every day you spend working on the permanency, you burn an appropriate level spell slot to cast that spell.
Additionally, you need to be somewhere while working on this where you can acquire all the extra spell components needed to stabilize the spell into a self-sustaining loop.
Finally, because of the delicate nature of permanency...you can't interrupt your work. If you're in the first half of the downtime and you skip a day...you get to start over. If you're in the second half of the downtime, you're operating at 3 levels of exhaustion and that day doesn't count towards your recovery time.
Whoever (or whatever) is the recipient of the Permanent Spell, must be present the entire time the spellcaster is working on making the spell permanent (so, the entire workday for the first half of the downtime). After that, they may wander off while the spellcaster recovers.