The first verb that comes to mind is rabaisser.

e.g.:

L’idée principale du Pervers Narcissique est de rabaisser sa victime pour la rendre moins importante que lui et rehausser du coup l’image qu’il a de lui-même

Also, déconsidérer, humilier, discréditer…

When talking about the attitude against those who try to differentiate themselves from the others, we can also use égalitarisme.

1/ There is no equivalent of “tall poppy syndrome” in French, I think, (“Syndrome du grand coquelicot” was rather funnily incomprehensible to read on Google) but there are partial ways to convey the idea: “couper toutes les têtes qui dépassent”, although that means more to force conformity rather than be jealous or resentful of the achievements of others.
You could look up “égalitarisme forcené”, but again, that doesn’t quite cover it: it is about not giving or recognising privilege, titles, or worship to people on the basis they distinguished themselves in a good way.

2/ Pull one down, down a peg, etc: I’d say the same as Jilagre: “rabaisser quelqu’un” is the first that comes to mind.

3/ crabs in a bucket: we have the same image used in an expression: un panier de crabes. But it is not so much about pulling back people who do better, it is more to describe a cut-throat environment, a group where everyone is mean to everyone, looking after number one, getting over each other’s head.
Funnily, the derived expression “ouvrir un panier de crabes” means more to open a can of worms: to trigger a situation that was contained (albeit in a bad way), a situation that is not good, and is going to be very difficult to rectify now it is triggered.

The difficulty find a translation to convey that resentment for people who do well is rather interesting culturally, isn’t it?

Two more references that you may want to look up, although not for actual translation :
the expression: Lit de Procuste (still quite used in France in the media and in literature)
the Nordic expression : “law of Jante”, which is Tall Poppy Syndrome pushed to the extreme, but very significant in Scandinavian countries.

And have a look at “méritocratie”: the fact that the word has a rather negative connotation in French, rather than positive, would indicate it carries a hint of “Tall Poppy Syndrome”…

French-speaking people complaining that they cannot perform as well as they could or should because of workplace harassment or continuous negative comments commonly refer to:

une culture / une politique de nivellement par le bas.

Which is to say that there seems to be a culture of using the lowest quality / productivity / general attitude as the benchmark everyone should aim to, lest they become ridiculed or have to face constant harassment. It is also possible to use a verbal form for it:

niveler par le bas

This expression is usually derogatory and intended at forcing the people responsible for this state of fact to rethink or at least adjust their position. It is probably best avoided, or a least mitigated in some way, in a neutral discourse. In an opinion-based discourse, even formal, it is commonly used, though the reasons for its use should in many cases be at least briefly presented.

For a more individual-based form of the same attitude (where the previous usually refers to a culture, a society of people), it is fairly common to hear (at least in Quebec):

(Quelqu’un) descend les autres pour se remonter.
    (Someone) brings the others down to gain some [social] height.

It is perhaps a bit wordy, but it makes use of the contrasting pair descendre-(re)monter, which perhaps explains its relative success.

It also contrasts the vile, needing to negatively reflect on others to acquire any type of lustre, to the excellent, whose value is recognized through the simple excellence of their acts, sayings or craftsmanship.

It also has a tinge of negativity, and its use is subject to the same remarks as the nivellement par le bas concerning the neutral discourse. It is mostly used in informal speech and gossiping.

“Rogner les ailes” Comes to mind as well.

To pull someone down would be rabaisser quelqu’un.

It can be used to describe a culture of pulling down people who are trying to get ahead:

Dans ce pays, on ne fait que rabaisser les meilleurs pour qu’ils soient au même niveau que les nuls.

It can also be used to describe an individual’s actions independently of the culture around them:

Les personnes peu sûres d’elles cachent leurs peurs derrière une attitude faussement confiante ayant pour but de rabaisser les autres.

The expression crabs in a bucket has a near-direct equivalent in French, c’est un panier de crabes, but that is only used to describe a situation of intense competition for a limited number of opportunities rather than an individual’s behavior.

La deuxième division de football est un vrai panier de crabes.

Cette élection est un panier de crabes.