Ok so I asked Sh. al Judai about a wide range of topics and their Islamic perspective. He told me some things to hold into generality:
The four major madhabs do not have significant differences on what they consider halal and haram when it comes to modern financial transactions, as they did not deal with the same things we dealt with today.
There are however minor differences on some points
Modern scholars, irrespective of madhab, take from a variety of sources (i.e., other madhabs and opinions, including those of non-traditional sources such as some trustworthy Shia sources (e.g. Sh. Muhammad Baqir al Sadr's book on banking has been cited by many) when it comes to Islamic finance.
There is a lot of Ikhtilaf between scholars on numerous points, coming from both a lack of specialization and a large requirement for Ijtihad.
These things are important because, people often ask what a specific madhab says about a certain transaction, but due to the nuances of Islamic jurisprudence today and the rapid rate of innovation in fian
Firstly, some definitions I learned (or improved upon):
Qimar (gambling): Gambling occurs when two parties enter into an agreement that based on an outcome related to probability or chance, one party will win and the other will lose (e.g. a coin toss with a winner and a loser).
Gharar (risk, in finance, risk arising from speculation, synonymous with uncertainty): Is in Islamic business terms, when the outcome of a transaction bears some level of uncertainty.
Investments with extreme Gharar are generally considered haram, for example, investing in something that is priced irrationally like a bubble with abnormal returns.
Mudaraba (speculation): When an investor/market participant holds a certain view of the future and enacts a transaction based on this view (e.g. buying stock in a company deemed profitable). [Note: the common definition people use for this is the idea of the Mudaraba contract, which is a very narrow element of the actual theory].
Mudaraba al Sila' (speculation on a good/service/commodity): When a market participant purchases something for a holding period in the hope its price will change in the direction they favor by the end of this holding period (e.g. buying a barrel of oil and keeping it for sale at a later date for a higher price).
Mudaraba al Si'ir (speculation on price): Where a market participant invests purely speculating on the price of something, and not on the thing itself (e.g., a contract for difference). This is haram as it is a form of gambling.
Bay' (sale): the exchange of goods or services.
Bay' al Salam (deferred sale): when a sale takes place, with the payment upfront and the good/service delivered at a later date.
In this sense, this type of sale can take place in two manners in modern terms:
Cash paid today for the provision of a good or service later;
A good or service is provided today for the provision of cash later.
Bay' al Inah (cash buyback): when a deferred sale takes place, but the good is sold back immediately at a lower price.
This has major ikhtilaf, the Shafi'i's say it is permissible, but the other madhabs say it is generally not.
All madhabs however agree that if the intention of the person is to essentially create an interest bearing loan in all but name (i.e. a heela [a work around]), it is haram.
On Bitcoin, he informed me that the UK Fatwa Committee (which consists of himself, Sh. Suhaib Hasan and several other prominent UK based scholars) had decided putting money into Bitcoin (and cryptocurrencies in general) were not halal at the current time for multiple reasons:
It is not currently recognized as legal tender in its current form, and several governments have instituted bans on it. This is contrary to the idea that Muslims should obey the laws of the countries they live in.
It acts as a source of extreme Gharar.
Whilst the idea itself does not go against Islamic principles (i.e., crypto in of itself isn't haram), the fact that it is primarily used for money laundering (whether by Chinese people going against capital controls or people who have illicit income through black market places), means it is currently unsuitable for use.
The Sh. corrected me on the point that speculation is not gambling Islamically, unless two parties are involved. I explained to him that, other than traditional retail brokerages which only generally sell vanilla stocks and bonds, most retail investors only have access to websites that provide CFD's (contract's for difference). He informed that this is equivalent to gambling if the speculation is being made in this case purely for the hope of an increase in price and contains two contracted parties.
However in the case of actually owning the commodity/bond/stock, without a contract for difference, there is no problem with speculation as long as it does not carry extremely high risk.
Most scholars today say insurance is haram, but Sh. al Judai said it is halal generally speaking and there have been modern scholars from all backgrounds who hold the same view. He says this because, Bay' al Salam can be considered a primitive form of insurance and has generally been allowed. The issue with insurance comes down to contract specifics and the type of insurance (naturally for example, life insurance would be particularly troublesome Islamically).
Insurance in itself can be sold in the form of an investment (e.g. an insurance company invests the money you give them for a particular policy in liquid assets, and grows a large enough balance sheet to pay off any claims) or it could be in the form of relying on the collective yearly payments from clients outweighing the collective yearly claims. The contracts themselves ultimately vary too and it really depends on a case by case basis.
I asked him a bunch of other things in my time and went of plenty of examples for my own project. I can share those too but I felt what I included here were things related to the questions in this thread in particular.