Is it possible to be Thane in multiple towns?

Yes you can be Thane in all Nine Holds in Skyrim simultaneously. You can also join all guilds at the same time. There are only two exceptions:

1. Imperial x Stormcloaks - you cannot be on both side of the civil war but whichever side you join will eventually conquer all nine holds and you can still be Thane, just under a different Jarl for holds conquered.
2. Dawngaurd x Volkihar Vampire - If you have the Dawnguard expansion you will have to choose between joining Dawnguards for their anti vampire equipments and crossbows or the Vampires for the Vampire Lord form.

Joining Dawnguards will not deny you of becoming a vampire after but you will not be able to get the Vampire exclusive Rings and Amulets plus your fellow Dawnguards will not welcome you during the period which you are "infected". Joining the Vampires however will make it impossible to get the Enhanced Dwarven Crossbow and it's elemental bolts, other dawnguard equipments will also be harder to obtain as you can only get them while you are fighting Dawnguards.

Your general question is "Can I actually be a part of every faction and get every achievement and all of that good stuff?"

The answer in nearly every Elder Scrolls game is a resounding "Yes". Skyrim, being more advanced, has some caveats and mutual exclusion situations which can happen.

1. There are opportunities to remove two different Daedric artefacts from play without receiving a second in return from their masters (Vaermina and Nocturnal). Doing so to both can make it difficult, if not impossible, to get the Oblivion Walker achievement. I recommend keeping Vaermina's.
2. The Stormcloaks and Empire remain neutral to you until you pick a side. Once you have done so, you are hostile to the opposing side, but can still enter their holds. I recommend not joining the war until you've completed the Alduin questline so that you can experience some of both sides, as well as enjoy the Season Unending quest which occurs midway through if the war isn't resolved.
3. It is possible to get kicked out of the College of Winterhold, so be a good student until you reach the top of the ladder.
4. The Thalmor are not joinable. They are technically "imperial" for most purposes, but are entirely intended to be an antagonist to both sides. They will behave as if you are an ally in the main quest, as well as the College of Winterhold quests. They can become individually and specifically hostile to you in the field (which can occur after you visit the Thalmor on one particular occasion). They have an Embassy near Solitude and a fortress on the coastline just behind that; entry into both leads to hostilities.
5. While you can technically join the Forsworn at one point, only one encampment of them in all of the Reach will recognize it. The rest remain hostile.
6. You can "join" the Penitus Oculatus in a manner by annihilating the Dark Brotherhood instead of joining them. There are no real benefits to this unless you're rolling a paragon imperial playthrough. They are just imperial special forces with no additional benefits, and you miss a lot of content if you do it.
7. If you become a vampire during the Dawnguard quests, and are siding with Dawnguard, they will have you cure yourself and tell you how before you continue, but you can be on both sides for a time. If you side with the Vampires, then Dawnguard will become hostile and your decisions are locked in. (Tip: Don't press Serana to cure herself; mention it, then wait till after Dawnguard wins if you sided with them, then press. Otherwise she won't do it. Also, being a vampire creates shortcuts when dealing with the vampire in Morthal).
8. The Companions don't care what you are or do. They do hold the key to being a werewolf, however, and without mods, you can't be both that and a vampire. This is normally due to disease resistance, but the vampires in Dawnguard can turn you anyway as an alternative to finding a Glenmoril witch head if you want to stop being a werewolf.
9. The Dark Brotherhood hits some high value Imperial targets. This will not change your allegiance if you are with the Imperials, but it can make it harder to finish the Dark Brotherhood if you are a Stormcloak.
10. The Blades will effectively break lore and refuse to work with you at one point if you don't kill one of your most trusted advisers and betray the Greybeards. These are meant to appear as opposing factions; one favors weaponizing the Dragonborn and the other favors a peaceful path. I highly recommend the mod "The Paarthurnax Dilemma", which lets you tell the Blades to shut up and remember you're basically Tiber Septim of the North. This will allow you to continue to recruit Blades, but remember, new Blades will stop being your companions.
11. Solstheim really only has one rivalry going, and that's a tribe of Nords versus a tribe of Reiklings. You do have to pick one side, and are considered a faction friend after that. This works like the Forsworn; siding with the Reiklings gives you a neat summon, but not all Reiklings in Solstheim will be friendly when you meet them.
12. You can't become High King, even if you get your hands on the crown. You also won't become the Emperor or a senior General in the Empire. You get a better rank with the Stormcloaks, but it doesn't mean anything. The Bard's College of Solitude doesn't have ranks other than anecdotal dialog mentions, the whole way you join the Forsworn is to free their leader and work for him, etc.
13. Thanes have limitations. If you are Thane, you can get away with minor crimes...once or twice. You can still go to jail. Don't get addicted to nobility if you have twitchy fingers around other rich people.
14. Also, if you become Thane of a hold and then betray that hold during the war, changing its allegiance and deposing its Jarl, you are sometimes effectively un-Thane'd though your holdings and housecarl remain with you and he/she still calls you Thane (I'm still not sure if this is a bug or as intended, as it only seems to happen sometimes. It happened to me in Whiterun and Riften). This means you can become Thane of that hold a second time. If you negotiate a town away during Season Unending and then take it back over for the original occupant, you don't have to re-Thane yourself twice. Once you are Thane in the eyes of one side, it stays that way.

If you don't like any of the above, there are mods which will allow you to be both (or a hybrid) werewolf and vampire, recover artefacts you didn't want to give up, change the effects of the war, etc.

Any factions not mentioned above, you can safely assume have no conflicts for joining them.

As for achievements, I recommend using the console to label your saves (if on PC) or splitting your savegames so you can restore to old saves if you are achievement farming and have to make binary decisions which exclude a path.

Yes you can. There aren't a lot of faction based restrictions in this game. Most of your actions will only affect the way NPCs greet you.

The biggest outlier would be the stormcloak-imperial-war. While progressing in this quest you will take different cities and the guards of the winner will replace the guards of the opposing factions cities. Again, this won't affect your game a lot.