Hope you are doing fine.
1)How do I register for trademark?
>>>>>>Some of our sellers get brand approved with trademark document also with Brand Authorization.There are two things you can do:
1)You can send your trademark document along with manufacturer Authorization letter to request amazon for remove any listing using your brand. In some case amazon still accepting it.
2)If any seller selling your brand of product and you own Trademark along with authorization letter asking to remove their listing. If required you may also warn them to sue.
If you have trademark and manufacturer authorization letter and some other sellers try to come into your listing then you can provide trademark document to Amazon which will lock all your listing for existing sellers.
When you apply for Trademark it takes 1 month or less than one month to come TM sign once your TM sign comes you can use this. This is only to protect your brand name. Once you got trademark with your brand then you can send that document to Amazon. For register sign R, it takes 6 month-1 years.
2)Will it be enough only registering brand with Amazon?
>>>Yes, getting trademark is enough to protect your brand.
3) How can I secure trademark name and logo? So that no other will claim it
>>>The brand name is unique. You can contact your CA for that and if you got trademark with your brand then no one can take your brand name and your listing will be locked. No one can hijack your listing.
Let me know if you need help with anything else!