Is it possible to have vinyl nearly noise free?

Yes it is possible. Vinyl is like wine. There are good ones and bad ones.
You usually get a near silent pressing with Decca, EMI, Erato, and Hamonia Mundi. Phillips and DGG about 1/2 the time. I'm sure there are other small European labels that do a good job. The worse pressings are from Rhino. I sent back 8 copies of Joy Division's Closer. Finally they ran out of copies and I picked a different record. And do not fall for that 180 gm thing. It does not get you a clean pressing. Analog Productions is the only American company you can count on. Mobile Fidelity occasionally. 

Once a record is noisy it is noisy for life. The secret to having clean records is don't let them get dirty in the first place. Use a dust cover and a grounded sweep arm (Sleeve City sells one for $20). Do not put stuff on your record. The placebo effect is rampant here. Either it all evaporates (the Freon in Last) or it gums up your stylus and glues dust to the record so the next time your stylus passes by it can be ground right in. 
The only reason to buy a record cleaner is if you buy used records. An ultrasonic machine using distill water is the best. I use a Spin Clean with distilled water only, usually to clean other peoples records. I rarely use a stylus brush because I do not put gunk on my records and the grounded sweep arm kills static and sweeps any incidental dust out of the way. Once a month or so I clean the stylus and the sweep arm with an artist's brush and 91% iso propyl alcohol. Mix it with 25% distilled water and you have the worlds greatest windshield cleaner!

There are numerous very quiet phono stages out there. Compared to the noise on the record their noise is inconsequential. Even the quietest record has a blowing sound to it. 

Whart, If iso propyl alcohol removed the cement from your stylus you need to buy a different cartridge from a different manufacturer. That is just a liability statement from Ortofon of which I have had several and have never damaged one yet. A lot of the stuff stuck to styli will not be removed by simple brushing especially just back to front. If some phono stages accentuate tics and noise then they have a frequency response/correction curve issue.
Just because one copy of a disc is quiet does not mean the whole run will be. They always get noisier towards the end of the run.
Tomic, Last is just freon. Freon is a group of halogenated hydrocarbons used usually as refrigerants and propellants. Freon is a great solvent 

Whoops, hit the green button by accident!

for poler molecules like oils and it evaporates very quickly. If you use the evaporation test you will notice that there is absolutely nothing in last that will not evaporate. 
Theo, the Sugar Cube is like having sex with a rubber on. It may be useful if the record has AIDs but it is not needed at all for a virgin record.

Cleeds,
 What's the matter? Don't like sex?

Whart,
I use the alcohol only about once a month and it is iso propyl not denatured. If denatured alcohol got back to the coils it could really screw things up. As a solvent iso propyl is just one click stronger than water. On a regular basis I just use a standard stylus brush which I use maybe once every third listening session. Because I use a grounded sweep arm and a dust cover my stylus does not collect much dust. What I really need to get is one of those USB microscopes. The monthly alcohol cleaning is based more on tradition then whether or not the stylus and cantilever are really dirty. It would be nice to be able to see it more clearly.

Whart, yes stuff will collect on the cantilever especially if the record is not grounded out. Again this is where the artist brush comes in handy. I use a very soft little pointed one. with alcohol you can safely clean the cantilever and the front of the stylus. 
What happened to your Airtight? I have a Supreme and it is built like a tank. It would take thousands of hours of play time to wear it out.
Atmasphere, I have never heard a phono stage make tics and pops. I would think that ultrasonic ringing would cause steady state distortion/overload if the unit's frequency response went that high. Granted my experience with multiple inexpensive phono stages is limited. My first preamp was a Dynaco PAS 3X and I was using a Pickering cartridge. I was listening through AR 2ax speakers which had a horribly muted high end so pops and tics were wonderfully suppressed. If the phono stage were at fault all records would make noise so if you have some very quiet records but others are noisy you can forget about the phono stage. Some cartridges seem quieter than others. It would seem stylus shape would have something to do with this. I have a friend who uses an old conical Denon because he has a lot of old less than optimally cared for records and he swears it is quieter. He was an early digital convert. He also ruined his CDs. Computer music is wonderful for people who don't take care of their stuff. This friend still has not got himself a back up hard drive! Busy Busy Busy.

Fleshler I always avoided playing an LP twice in a day as the prevailing thinking was as you describe. It is true the vinyl under the stylus is under immense pressure and is described as going liquid (never proven as far as I can tell). The mass of vinyl involved is so small that the surrounding vinyl acts as a heat sink and the temperature returns to normal immediately. I have a tire infrared temperature gauge and can not detect any change in the temperature of the record while playing. I think as long as the record and stylus are clean it is safe to play a record over again if you have too. But who wants to listen to a record twice in a day;-) 
"Last" is a joke. It is nothing but freon. You can prove this to yourself easily. With a dropper place a drop of Last on a glass slide and let it evaporate. Put another drop in the same place on the slide and let it evaporate. You can keep doing this a thousand times over and you will never see anything remaining on the slide. Nothing, Nada, Zero. That is how gullible we are. Will freon change the vinyl in any way? Absolutely not. It is totally inert which is one of the reasons it is a great refrigerant.
Lps are Lps. Some are quiet, some not. The best way to keep a quiet collection is do not buy used records. There are exceptions for instance if you are buying the estate of an expired audiophile who had so many records he may have played one once a decade. Make sure he did not buy used records.