Concerning question 1: for free Burnside groups of odd exponent $n\geq 665$ the decidability was shown by S.I.Adian. Lysionok proved the same in the case of even exponent
$n=16k\geq 8000$. The corresponding deciding procedure is just Dehn's algorithm.
It will be interesting to know how to solve the word problem for groups $B(m,n)$ when $n=2k$, where $k\geqslant 665$ and odd. These groups are infinite (it follows from the cited result of Adian), but it seems that the decidability of the word problem for these groups is an open question.
As to the question 2:
The pure existence of non-finitely presented group with undecidable w.p. easily follows from the result of S.I.Adian who proved that there are continuum non-isomorphic periodic groups with fixed number of generators $m\geqslant 2$ satisfying the periodic law $X^n=1$ if $n$ is odd and $n \geq 665$.
One should mention, that the number of "additional relations" (besides all periodic) is not necessarily finite.
Later M.Sapir constructed an example of periodic group with finite number of additional relations. The paper is "On the word problem in periodic group varieties", IJAC, 1991, V.1, No.1.