Ummm... it might be normal to try, I can't answer that part, but I can tell you that it is NOT normal for the parent to allow this.

No, yet SO's DS11 acts just like that! That's how a 2-3 year old acts and we as parents correct that behavior so at 7-8 they aren't acting like that. Unless the child has learning disabilities or other issues then maybe. But a normal 7-8 that was raised properly to not act like a feral animal.

Not really, no. Although I suppose it depends on the kid & how they raised. My SD may do something to this effect once in a while, to push buttons, but most kids learn manners along the way & don't behave that way regularly by 7-8. Manners are important in step situations & in life. No ones parents will want a wild animal in their home, meaning his friends may not invite him over if this sort of thing continues. Sad really.

You are going through what I went through the first year I was with DP.
NEITHER PARENT TAUGHT THE KIDS CORRECT BEHAVIOUR!
As a childless, by choice, person I was horrified and wanted to correct things ASAP. But that backfired big time. I'll take your questions one at a time since I've been through this with 2 SDs, one then 7 and one then 9 (now 10 & 12 and much better).

Is it normal for a 7 year old to walk on top of a living room table with their shoes on:
The SDs treated furniture as playground/gym equipment. DP (and BM I found out later) allowed this. I could do little to stop it if I was at DPs home (all rental furniture that came with the house) but I didn't allow it at my apartment. And once we bought a home, I didn't allow it with my furniture. I spoke up if it looked like SDs were going to hurt themselves or someone else - or if we were in a hotel room. If he wanted to lose his deposit that was his choice in his rental. He started speaking up when they began actually breaking the furniture - this includes curtains, blinds and towel bars. Pick your battles.

...or get on his knees in a restaurant in a booth and look and play around at the people sitting behind us or climb under the table in a restaurant:
I was HORRIFIED when I went out with DP and SDs the first couple of times. It, like their home, was treated like a playfield. I was not quiet when we were in public and threatened to leave the restaurant unless they settled down (and I meant it). There was to be no more eating the sugar, drinking the creamer, jumping on/kicking seats, or having "races" in the aisles. If they were done eating before the adults were done, they were to sit and wait. Hands were washed before dinner and dirty coats taken off. Treat the waitstaff politely. If this fails, and it failed at first for me, STOP GOING OUT WITH THEM. I went about a year of refusing to go out to eat with them. It worked.

...or to jump up in down in a infant car seat:
NO MESSING AROUND WHILE IN THE CAR - PERIOD.
Foot down. Pull the car over. If you are not driving, and your spouse refuses to pull over, then refuse to ride with them as a group unless you drive.

...and not know what a penny is:
Skids not being taught the value of money is something you need to ignore.
Not your issue. Stay out of all money talks. Don't give it, don't lend it, don't buy anything for them.

...or how to tie his shoe?
I think this is actually common with younger kids these days - most kids shoes are strap/velcro on. Neither SD wanted to learn to tie shoes until they got a little older. Both think shoes (mostly sneakers - sandals are ok) with velcro are for babies. Kids will let you know when they want to learn to tie, both SDs wanted to around ages 9-10 or a little sooner.

The question SHOULD be "is it normal for parents to ALLOW this type of behaviour?"

No it is not.
Is it normal for kids to try this and attempt to get away with it?

Yes it is...for the FIRST time only.

All three of Chef's kids had this type of atrocious behaviour to the point of hate stares at ME from the other patrons. Chef ALLOWED it because he "didn't want to spend his short visitation time (EVERY weekend) instructing/training/guiding/disciplining his kids" (TM)

Especially when he was competing with the Girhippo (BM) for who could do the LEAST amount of parenting

RED FLAG OF SHITTY PARENTING!

Feral skids..... The guy I am with now has SD8 and she has horrid table manners. Whines like a baby/ throws a tantrum if she doesn't get her way, chews with her mouth open , wipes her hands on her shirt, makes a big mess all over the table, lies down in the booth, won't sit still.... I HATE going out to eat with them. So embarrassing ! My SO knows how I feel and he is working to correct the kids bad behaviors.

The behavior your DH is allowing is not normal. He is not doing his kid any favors by allowing him to act like an animal. Good table manners are a basic social skill.

for a child of seven? no.

for a toddler, yes. and the the toddler should be told to sit down and be polite. adult sd has atrocious behavior at house parties where she helps herself to the food before the other guests have been invited to the buffet table. embarrassing, but dh thinks of her as baby still and its not problem. she looks like a rude glutton.

Got an 8 year old here and although she might enjoy doing those things she knows better. She also knows how to tie her laces but only because I taught her how neither of her parents had.

I had lunch today with a just turned 9 year old who is a COD. She ordered fried crawfish tails and when they arrived she very politely asked me if she should use her fork or would it be okay to eat them with her fingers. I said "either". She tried the fork first and then switched to making it finger food.

She did not dance on tables or act the fool. And yes she is a COD whose father has at times over indulged her. But underneath it all she remains a polite child who wants to do the right thing.

So not all indulgent fathers create feral children. Fortunately her dad (who has 50/50 custody) also wants her to grow up to be a decent person.

No. That's the behavior of a much younger child. If that behavior is still going on at seven I would say there's been some very lackluster parenting