Just a couple tools come to mind..

Metric sockets & wrenches
Torx bits - a lot of these on the car
Rear brake caliper retraction tool - if you plan to service the rear brakes

Other than that, most standard tools should be fine.

There are a lot of specialized tools if you go deeper into servicing, but those are typically expensive and probably not worth the investment for one time use...

Thank you all for your feedback. I did take the care to a local specialized mechanic (the same one who replaced my clutch/flywheel) and they confirmed half the list advised by the dealer was unnecessary (had my suspicions and I do greatly appreciate you guys and the mechanic confirming).

That being said, most of my mechanical experience is on older vehicles of the American breed. If you guys have suggestions as to some of the tools needed for the simpler, more basic repairs of this Mini, and advise on the proper approaches, I will take them as well!

Side note: Anyone in the NC area I would suggest Premier1 Auto of Apex for major mechanics/performance tune-ups on import vehicles. I used them previously when owing a VW Jetta and highly recommend them.

Great response from Dan.

It is true that many all wheel drive models are sensitive to using tires of different overall diameters which causes the tires to spin at different speeds albeit slightly. This can be enough to overheat the clutch pack in the differential and cause damage.

I am not sure that letting the new fluid settle in and replacing the tire will resolve the issue - but if the shop says it will, that may be a lower cost solution to try...

Let us know how this turns out.

Well said Dan.
We use the dealer for the warranty period only, then it's either me or an independent shop.
I've had my issues with dealers....say no more.

As with any used car, it is up to the owner to decide when enough is enough.
Individually, the prices you quoted may not be too bad. $1200 for tires, installation and alignment seems reasonable for any car. Replacing an exhaust clamp at $125 at a dealer is probably on par, though you could do it yourself if you have the tools, the skills and the inclination to do it. Ditto the drive belt and upper coolant hose, though they may be trickier. By "oil housing gasket" you mean the oil pan and that would depend on how complicated it is to do if other components (maybe exhaust, drive train components etc.) need to be removed to gain access.

If it were me, I'd start with getting quotes for the necessary things from a non-dealer garage familiar with the model of car. Then decide which things are urgent, such as the upper coolant hose, the oil leak and the drive belt. I'd go ahead with the tires as they may solve or alleviate the drive train problems.
Why a non-dealer? Dealers generally tend to advise you of everything wrong and urge you to get them fixed ASAP. They make their money in their service departments and they have much higher overhead costs and higher labour rates. On the other hand, an independent garage has lower overhead, better labour rates, and most importantly need to maintain a good reputation. They will give a good customer a break on some repairs if possible, say combining repair events to save on the total shop time cost. (Dealers quote repair times individually from a "book" and don't typically charge less if they can do it in less shop time.) An independent garage will advise you if a repair (e.g. a minor oil leak) is urgent or not. A dealer won't.

One thing to consider is the availability of parts. Most auto makers are limiting their supply of parts to a 10 year supply. For example We bought a used 2010 Jeep in very good condition and last year learned that the fuel tank support had rusted and needed replacing. We deal with an independent garage. They found that after 10 years, they could not get a replacement new part anywhere in North America. They searched the used part network and came up empty as well. They ended up rebuilding it by cutting out old rusty sections and welding in new sections, painting and undercoating it. Despite the time they had it (about a week) and the work and material that went into it, the total bill was less than half what we expected. There was no estimate - we can trust them. Did I mention they are within walking distance of our home?

Others with BMW Mini experience can chime in and advise on the specific issues.

Question for you guys (this will be a bit of a read so please hang tight lol). I picked up a 2011 Mini Cooper Countryman S All4. The rear differential has been acting up due to improper care of the tires (basically only getting 1 to 2 tires replaced when needed vs. all 4 and keeping the tread balanced). My understanding is this can cause extensive wear on the rear differential/transfer case. 

It's been estimated at a maximum of 36k miles this car has been driven around with all 4 tires varying (which leads to the current situation with the differential).

Recently I had the differential fluid and transfer fluid flushed out, replaced, and some high friction solvent added. The next step would be getting all new tires, balanced, and an alignment. 

Now, I've had one dealership say that I need to replace the whole clutch pack (along with other various things totaling up to 12k) and I've had another place (who specializes in imports) tell me the issue may resolve itself once the fluids settle in and new tires are thrown on, and they can replace the other concerns at almost half price.

Here are the questions: Is it worth putting in some work and keeping this car? Should I be concerned about having to replace the drive terrain assembly? Or will the issue balance back out with new tires/alignment? I would love to keep this car IF I can fix things at a minimum while knowing the car is getting proper care (I do strongly believe in proper car care, but don't want to be lead astray on someone else's desire to swindle the little guy). 

Issues include: Replacing oil housing gasket ($1,200); Replacing clutch pack (if needed $8,000); Replacing tires + alignment ($1,200); Replacing drive belt ($200); Replacing Exhaust clamp ($125); Replacing upper coolant hose ($245)

Variables: I will need the car for long-distance travels on regular basis; Limited finances; Plan to be traveling much with my kids (or to see my kids) 

Thank you guys in advance for your help!!!