Was TIM involved in the Andromeda Initiative....?

Nah, the airlock will meet her only if she is still loyal to Cerberus/their ideas. And yes, we most likely won't be able to throw her out, so the airlock will be a metaphorical one. 

But seriously now, I hope that, as @hanshotfirst said, the possible role of Cerberus is limited to Cora having some kind of daddy issues because, as much as daddy issues has been overused in the past, nothing beats ME3 Cerberus in that department.

For once I hope it's not Cerberus.

I mean it's pretty telling that everyone, Bioware included, is instictively pointing towards Cerberus as a clarification for any conspiracies or otherwise unexplainable events that happen in the Mass Effect univese.

Can't anything happen in this series without Cerberus having finger in it? Almost all of the comics have been about Cerberus, 3/4 of the novels are about Cerberus. 2/3 of the games feature Cerberus and whole lot dlc and spin-off works feature Cerberus as well.

This.

We had way too much Cerberus than was needed already. By the end of the series they had started to overshadow the Reapers, particularly if you include all the tie-in novels and comics which as you noted were almost exclusively focused on Cerberus.

Cerberus, in short, became a writer's pet. Let them be dead and gone with ME3.

I wouldn't mind Cora having familial ties to TIM, so long as she's in Andromeda to get away from him and isn't leading a Cerberus cell about the arks. If we need shady human characters with a secret agenda, let them be something other than Cerberus.

People like to say that Liara is the writer's pet of the trilogy. Please, Liara has nothing on Cerberus.

Wasn't his father's last name Sanders, and the no relation was with Kahlee, or am I mixing up two different conversations?

It's just as likely that this is a nod to Andromeda the TV series. The chief engineer on the show was named Harper.

The first officer on the ship was also played by Lisa Ryder.

Honestly, I'm leaning more towards this as Bioware has been very intentional about severing ties to the old trilogy. They specifically set the launch of the Andromeda mission between ME1 and ME2 so that they wouldn't have to deal with the messy continuity that ME2/ME3 brings. A squad member having ties to a character from games that won't have any reference points in Andromeda (meaning, the mission launched before the events of ME2, so they'll have no idea what happened) just seems odd.

Very unlikely, this game's story takes place after and separated from the events of the trilogy, and i hope TIM has nohing to do with the iniciative, for i don't want cerberus in Andromeda
if the trilogy's story ended, then all character's setories ended aswell

I want Andromeda to be completely independent from the trilogy. I just want the pain to go away and never come back.

Well, from the timeline of the series, Shepard wasn't ressurected before 2185, the same year the Andromeda Initiative was launched (http://masseffect.wikia.com/wiki/Andromeda_Initiative) . This probably means that TIM was busy with Shepard and the Reaper's threat, not to count all the politics and that ninja (Kai Leng) and thus wouldn't be involved in the initiative, at least not in a deeper way. He could be the mind behind Jien Garson, since the initiative was founded years before the events of Me2 and 3.
AI being a multi-species project, they would never allow TIM, Cerberus's Mastermind, a known alleged terrorist and pro-human organization , to be part of this project. Specially considering that when the Initiative was founded in 2176, the humans existence were merely known, since the first contact war, which started on 2157.

Considering all of that, if TIM is somehow involved, it certainly would be in a hideous way, but it totally makes sense for him to be involved, since the project was being born when Cerberus was in its apex so to speak. TIM would surely know about the project and, considering he is always acting so the humans become the masters of the universe, it would be an obvious move for him to somehow influence this "another galaxy" project.

You have answered the question: TIM would never believe in losing.

A more serious answer: Mac Walters has denied any Cerberus connection and I believe that this is no marketing speach. Even if Cora Harper turns out to be TIM's daughter, running away from his cruel father might be her reasoning for joining up. Back to TIM: he doesn't strike as a leader who sends people out on missions where he won't see the results. So it doesn't make any story sense, which is reason #2 besides the negative feedback Walters got on Cerberus in ME3.

Vega knows the illusive man's real surname?

Same here. I hope she has no connection to him, but the last I want is to see even one more dumb Cerberus soldier. So if she is his daughter, running away would be less painful for me. I am relatively optimistic though that Mac has put Cerberus to bed, because they were only needed as an enemy because we couldn't fight Reapers on foot too often.