Can a wallet/exchange provider ask for transaction details

Am I obligated to share transaction details with my wallet/exchange provider for any payment I made to an individual, like subscription to a website?

A proper noncustodial wallet will impose no such restrictions.

A business that is running a wallet service might impose restrictions though. So this question really just depends on the terms of service / policy of that business. If they are holding your private keys, you will be 'obligated' to follow their rules, though I'm not sure I agree with the use of that word, maybe the word 'forced' would be a better pick. Not your keys, not your coins, and all that.

Irrespective of the above point, If I pay for a subscription using Bitcoins to someone, and the recieving party misuse the coins for something illegal unknown to me, will I be in problem?

Most businesses (ie exchanges) that impose such restrictions do so for legal reasons: the jurisdiction they operate within requires some sort of KYC/AML procedures. So these businesses will often employ the use of a chain analysis service, which can supply the business with more info about happenings on the blockchain, thus satisfying regulators. Chain analysis services are not capable of perfectly knowing what on-chain transactions represent, but they can make guesses based on certain heuristics, and data gathered from various sources.

Some easy ways to mitigate against this risk:

• Don't store coins with custodians that might restrict your access
• Create another wallet, send coins to it, perform a self-send or two with that wallet, and then finally, pay the coins to your intended destination. This adds some extra steps between the address which is known to be owned by you, and the person you are sending payment to, breaking the obvious link between you two.
• Use a wallet which creates coinjoins to obfuscate your UTXO history, such as Wasabi, or JoinMarket
• Use a wallet which has payjoin/P2EP enabled (this is currently an ongoing development)

There is a lot of nuance to bitcoin privacy, I'd recommend giving this wiki entry about Bitcoin privacy a read if you're interested.

Generally, the sort of worry you have doesn't seem to cause problems for users these days (I haven't seen many complaints of this online), but if you're worried I would simply suggest adding a new wallet and a transaction or two in between the regulated custodian, and the outgoing payment.

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More broadly, in order for Bitcoin to really be useful, users need to not worry about these sorts of risks. Bitcoin can function in a way which allows all transactions to be perfectly traceable, but it can also function in a way which provides the user with financial privacy. As a user, you can employ certain techniques and strategies to help retain your privacy, but remember also: the businesses you support are the ones which will continue to grow in this industry. If you don't like the sort of existential risk imposed on the network by KYC'd exchanges, don't use them. As they say: vote with your dollars (or in this case, your sats).