"The first plate"
Translation:Der erste Teller
Because after definite articles we need to use weak inflection for adjectives http://en.wikipedia.org/wiki/German_adjectives#Weak_inflection
and since it should be Nominative case masculine gender the ending is "-e".
It seems very complicated at first but as you notice that you "only" need to learn N and A case endings for mixed and weak inflection (because all others endings are -en :) and when to use which inflection it gets a bit easier :) Strong inflection endings are almost the same as those for definite article. (Genitive case for m and nt is different)
Yes, if there is no article (indefinite nor definite) before an adjective then we use strong inflection. Of course there are some other rules regarding the usage of strong inflection (check the link I posted before) but the main one is the one we mentioned ("when no article is used").
Technically you conjugate verbs and decline adjectives.. but I understand your question. Here "erste" is an adjective so it declines exactly like an adjective.
"Der Erste" here it's a noun meaning "the first one". It takes the same gender as the implied noun (very confusing for English speakers to have to conform the Gender of an unspoken noun). The ending of Erste changes as well, but that's for a future lesson.