Can mother submit her copy of her passport for documents that need to be submitted after the non immigrant visa has been paid?

Yes, submit mother's passport with an explanation about why the father's is not being sent.

Informing USCIS (to update I-485) about my newborn

It is permissible to update your information at the time of your interview.

How long it takes to the I130 petition by a permanent resident be approved?

It takes about 3-6 months to get an approval.

Submitted I-130 and I-485 concurrently, went for interview with my spouse yesterday. Interview was great.

Based on your post, you should receive the Green Card in about three weeks.

Expect the receipt within a few weeks and a response on I-130 in about three to six months.

Can I travel to Spain with a conditional residence?

No problem traveling to Spain in April.

Is there anything else that I can do to understand why it is taking longer? My father's case i485 approval has not been updated?

These kind of events are not unusual and of course well within the 120 days they are allowed. Do not panic, and enjoy the fact Mom was done.

What do I need to cross the US border if I'm visiting Canada or Mexico?

Your passport and visa are all that is needed.

Is it possible to hire a new attorney from another law firm to accompany me for the interview?

It is legal and permissible to choose who you wish to represent you.

Does that office qualifies as a residency since the lease was on my name August 2014 - August 2015 period?

No an office is not considered a residency.