No, Stripe take their cut and the tax implications are down to the seller. However! They did recently acquire TaxJar (https://stripe.com/newsroom/news/taxjar) so I imagine things will become easier in this regard, reporting and filing, soon.
That's an interesting bit of news, in that at least it confirms Stripe have recognised the danger. I will be happy to be proven wrong about this, but I fear their reaction is too little and too late for merchants here in the UK. The likes of Paddle are already offering much simpler tax arrangements and other significant advantages as well, and as you would expect, this is already disrupting some markets that might have been considered Stripe's natural territory just a few years ago.
Stripe is ONLY a payment gateway, 'Stripe handling taxes' is a complete myth that is always brought up by uninformed US Stripe customers only to hype Stripe up.
Stripe customers in the EU have to deal with this VAT tax pain already which is why they either don't use Stripe and use Paddle instead.
Not really. That sort of payment processor might allow you to specify a tax rate to include on a bill and maybe to configure different rates that are automatically selected depending on where your customer is. However, typically they do not actively monitor and update the current tax rates for different areas, nor do they handle the reporting and remittance to all required authorities, so they only take care of a small part of the overall compliance requirements.
There is another type of service becoming increasingly prominent, which is a "merchant of record". In effect, the service becomes a reseller for your product or service, and since they are then the vendor for legal purposes, they also become responsible for all the end customer taxation issues. You in turn typically deal with them as a much simpler B2B relationship between just two entities.