Having worked in the domain name industry since 2001, I would always suggest a trademark search before acquiring names. Please keep in mind that when registering a name, you agree to the Uniform Domain-Name Dispute-Resolution Policy (UDRP) as part of your registration contract. This means that you agree not to infringe on rights of others with the registration of your domain.

When selling a domain name, it really comes down to your contract/agreement with the buyer. Depending on the details you might be held liable for any potential trademark infringements at a later time.

DIsclaimer: I am not a lawyer, it would be best to consult with a lawyer.

No. Unless you have an agreement with the buyer in which it states that you have to.
That said, if you are aware of a trademark issue when making the sale, and a problem arises for the new buyer, they might claim that you acted in bad faith and request compensation for damages and lost profits (but they would have to prove that you knew).
I have successfully helped over 300 entrepreneurs and would be happy to help you (after scheduling a call, please send me a message with all the background information so that I can prepare in advance). Good luck

No. You don't need to do this extra work before you sell them.

Though it is the buyer's primary responsibility to verify and validate for trademark infringements, however this does not waive all your liabilities.
If the domain names are violating any trademarks before sale, then you would be accountable too for that extent
You will be liable for damages and claims for any infringements till the time of the sale

It absolutely IS your responsibility for you do a trademark search before you sell a domain name. Otherwise you could be selling someone else trademark property which you have no right to do. Leading to lawsuits and a damaged reputation. Research before you claim and sell and domain.

No.

The responsible party is the person placing content on the site.

Also, there is no trademark violation, so long as...

1) Person using the domain was using the term before someone else trade marked the name/phrase.

2) The domain relates to a different niche.

For example if the trade mark is "Best Stuff" for a soap.

Then the domain uses "Best Stuff" for Vegan recipes, there is no trademark infringement.

If you have questions, best to ask some expert about this.

Tip: Also, if you have Legal Shield coverage, very few people will ever have the budget to take you on in court.

Search Clarity for other questions involving Legal Shield for details about this service.

Definitely you shouldn't register domains in bad faith to exploit someone's pre-existing brand name. That would be cybersquatting, and it's prohibited. The UDRP can result in the domain being confiscated on that basis.

That said, just because someone uses a brand name similar to your domain does NOT mean that they necessarily have any right to own – or prevent others from using – that domain. It's commonplace to see many companies sharing similar or identical names. And when the domain registration predates someone else's usage of a name, that other person generally has no rights whatsoever to the domain.

Trademarks are tied to a (relatively small) field of usage. So many different companies might share the same brand name but use it in non-overlapping applications or jurisdictions. And in doing so, they generally don't violate one another's trademark rights. In fact, the USPTO (which governs trademarks in the USA) often grants trademarks to the same brand name to multiple different entities simultaneously.

If you're selling a domain name, then it is the BUYER who wil be using it. And it's that buyer's usage that determines whether the domain would violate someone else's trademark or not.

For example, if "Tooth Fairy" is the name of a dentist's office with a trademark filed already, and your buyer intends to use "Tooth Fairy" in the same way, then they will violate an existing trademark. But if they intend to use "Tooth Fairy" to brand a video game, then they will not.

Usage is the buyer's affair. It's beyond your control as the domain seller what the buyer will do with the domain.

That said, if you are registering domains that contain – let's say – "Google", then that's clearly bad faith. Only register domains where there are legitimate non-infringing uses possible or many divergent potential buyers, no single one of whom can lay sole claim to the domain.