[collections] Is there a typescript List<> and/or Map<> class/library?

It's very easy to write that yourself, and that way you have more control over things.. As the other answers say, TypeScript is not aimed at adding runtime types or functionality.

Map:

class Map<T> {
    private items: { [key: string]: T };

    constructor() {
        this.items = {};
    }

    add(key: string, value: T): void {
        this.items[key] = value;
    }

    has(key: string): boolean {
        return key in this.items;
    }

    get(key: string): T {
        return this.items[key];
    }
}

List:

class List<T> {
    private items: Array<T>;

    constructor() {
        this.items = [];
    }

    size(): number {
        return this.items.length;
    }

    add(value: T): void {
        this.items.push(value);
    }

    get(index: number): T {
        return this.items[index];
    }
}

I haven't tested (or even tried to compile) this code, but it should give you a starting point.. you can of course then change what ever you want and add the functionality that YOU need...

As for your "special needs" from the List, I see no reason why to implement a linked list, since the javascript array lets you add and remove items.
Here's a modified version of the List to handle the get prev/next from the element itself:

class ListItem<T> {
    private list: List<T>;
    private index: number;

    public value: T;

    constructor(list: List<T>, value: T, index: number) {
        this.list = list;
        this.index = index;
        this.value = value;
    }

    prev(): ListItem<T> {
        return this.list.get(this.index - 1);
    }

    next(): ListItem<T> {
        return this.list.get(this.index + 1);   
    }
}

class List<T> {
    private items: Array<ListItem<T>>;

    constructor() {
        this.items = [];
    }

    size(): number {
        return this.items.length;
    }

    add(value: T): void {
        this.items.push(new ListItem<T>(this, value, this.size()));
    }

    get(index: number): ListItem<T> {
        return this.items[index];
    }
}

Here too you're looking at untested code..

Hope this helps.

Edit - as this answer still gets some attention

Javascript has a native Map object so there's no need to create your own:

let map = new Map();
map.set("key1", "value1");
console.log(map.get("key1")); // value1