One-way telepathic link only
Dominate person's telepathic link is only used for issuing commands to it not for communication
In 5e, spells only do what they specifically allow you to do, nothing more or less. And the way the spell is written only allows for a couple specific uses of the telepathic link described in it.
Dominate person gives you a telepathic link to the dominated creature that you can use to issue commands to it which it then does its best to obey.
While the target is charmed, you have a telepathic link with it as long as the two of you are on the same plane of existence. You can use this telepathic link to issue commands to the creature while you are conscious (no action required), which it does its best to obey.
The spell then proceeds to describe how this telepathic link can be used to issue commands to control the creature in either a broad or a specific way.
Nowhere in the description does it say that any other kinds of communication can take place through that telepathic link. Only that the link is created and that it is used for issuing commands. And, if the spell does not specifically give you an ability, then you do not get that ability.
Jeremy Crawford has specifically ruled on similar cases several times (here, here, and here for example) and each time the answer says that the ability must say that the creature is able to respond or else they cannot.
In the end, the purpose of dominate person is to control a creature, not to talk with it.
Spells that allow communication specifically say it and outline the rules for it
For example, find familiar says:
While your familiar is within 100 feet of you, you can communicate with it telepathically.
Note that the wording is different between the two: dominate person says "you have a telepathic link with it" whereas find familiar says "communicate with it telepathically". Communicate makes it perfectly clear that the telepathic bond is used for communication. It then proceeds to outline how that works:
Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.
Note that this is a major and relatively unique feature of the find familiar spell to be able to see through the familiar's senses and it comes with rules and consequences. If the designers wanted to have this be part of the dominate person spell they would have HAD to have added some rules to explain how it works. This is much too major a point to have just overlooked since it changes the spell entirely. Thus, it seems they did not intend for the spell to work that way.
This makes complete sense because the link created by dominate person is not for communicating, but only for issuing commands. On the other hand, find familiar is specifically enabling for two-way communication between caster and familiar.
Jeremy Crawford has also supported the idea that telepathic abilities can be one or two way:
Some abilities allow sending thoughts, some allow reading thoughts, and some allow both.
If you want to communicate telepathically, you have to use telepathy (or something like it)
You create a telepathic link between yourself and a willing creature with which you are familiar...Until the spell ends, you and the target can instantaneously share words, images, sounds, and other sensory messages with one another through the link
It is important to note, that without the above text, the spell would do nothing. A telepathic link without a description as to what the link does is useless. Clearly, though, telepathy is intended to be used to allow two creatures to communicate mind-to-mind and so it describes how that works and the rules associated with it. As discussed above, dominate person's purpose is different and thus it is described differently. Since dominate person doesn't have any of this kind of text it is clear that the telepathic link described therein is different and not intended to be used the same way.
Your specific questions:
If the link is two-way, can the target communicate (willingly or when
No the communication is not two-way. No information can be exchanged. The ability to communicate is something that would have to be written into the spell (and is for other spells). Also, this interpretation would make dominate person much more powerful then higher level spells which makes no sense.
Does the target need to spend a turn or bonus action when using this
Target cannot use the link (see above).
Does the caster perceive what the target perceive like Find Familiar
No, that effect is specific to find familiar and there is no similar language in dominate person.