UCEED is the entrance exam if you wish to pursue design. Check details here https://engineering.careers360.com/articles/uceed. The next step would be CEED which is the PG entrance exam (https://engineering.careers360.com/articles/ceed). However, you can write CEED even if you are an engineer. Either way is fine to pursue specialization in product design.

No , you can apply for bachelor of design.