You are mostly correct; except in your understanding of the wilds.
To determine if you are allowed to pick up the (entire) discard pile; follow these rules:
If the discard pile is frozen, you can only pick up the discard pile if you can immediately meld that top card along with 2 more of the same card from your hand. A wild cannot help you here; you need 2 natural matches, not including wilds. If you have never melded before; you need to also meet the minimum meld value (by melding whatever else you want from your hand at the same time). You can count the top card in the points for part of that minimum meld; but you cannot count anything else in the discard pile.
If the discard pile is not frozen, you can only pick up the discard pile if you can legally meld the top card of the discard pile. This could be melding it by just adding it to an existing meld; or melding it along with more cards from your hand (which could include wilds). There is no minimum meld to worry about, because if the discard pile is not frozen, you have already melded.
In either of the 2 cases, if you pick up the top card of the discard pile, you also pick up the entire pile.
The discard pile is frozen (for your team) if either there is a wild in the discard pile, or if your team has not yet melded at all.