Passport and Visa Query. Please help.?

Hi, I recived ILR (Indefinite leave to remain) in the Uk about a year. After that I have applied for a banladeshi passport since I am a bangladeshi citizen. Can I travel to bangladesh and back with the bangaldeshi passport that ive got after reciving IRL or do i have to carry the ILR with me when I travel.

I'm also going to Camp america this summer, if i get a american visa to travel there do i have to carry the british ILR with me when im traveling. Is there any way I can get the ILr transfered top to my bangladeshi passport.

Finaaly, Iam also applying for a TRavel odccument. there is four types

-Refugee Convention travel document
-Stateless Stateless person’s document
-You want to leave and not come back One-way document (also known as an IS137)
-Refused a passport by your own national authorities Certificate of travel

Which type of travel doccuem should i apply for to considering the fact I have uindefinate leave to remain in the Uk and want to to travel back and forth from other countries..

Please help, any relevent info is greatly appreciated.