What place do ritual and ceremony have in the church today? (Non-Liturgical POV)

Is the ceremonial aspect of all the rituals laid out in the Sinaitic law, the specific guidelines for priests and sacrifices, and the ceremonial and symbolic nature of communion and baptism relevant or important?

Okay, the ritual you're referring to here:

The ordination of Aaron is a highly ritualized procedure. The sacrifices are each a little different and are laid out as detailed rituals performed by priest and worshiper. Even the camp around the tabernacle was highly organized and only set out when the trumpets were blown a certain way.

In my opinion is a type of Jesus and the two thieves. The explanation of this is far too long for this answer, but there another person out there that I agree with wholly at this point, you can find that here.

Therefore, with that being the interpretation of the ritual we can see that this ritual has no more significance because a sacrifice is no longer necessary for our salvation, a pure and Holy sacrifice was made.

Now, I would like to focus on the following text for a minute:

...and the ceremonial and symbolic nature of communion and baptism...

These two ceremonies are still very relevant and in fact 100% necessary to our salvation. Let's break each one of them down here.

Baptism

Let's start here:

18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit: 19 By which also he went and preached unto the spirits in prison; 20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water. 21 The like figure whereunto even baptism doth also now save us (not the putting away of the filth of the flesh, but the answer of a good conscience toward God,) by the resurrection of Jesus Christ: (1 Peter 3: 18-21 KJV)

Clearly baptism is still very necessary for our salvation. But let's not just concrete this on one account, let's get another.

15 And he said unto them, Go ye into all the world, and preach the gospel to every creature. 16 He that believeth and is baptized shall be saved; but he that believeth not shall be damned. (Mark 16:15-16 KJV)

So, now we have both Peter and Jesus speaking about the necessity of baptism for our salvation. But how do we do it right? I mean there must be some instruction. Let's start with Jesus:

18 And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. 19 Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: (Matthew 28:18-19 KJV)

Alright, but what is the name of the Father, Son, and Holy Ghost? Let's start with the Father:

I am come in my Father's name, and ye receive me not: if another shall come in his own name, him ye will receive. (John 5:43 KJV)

Now, I don't think we need to review the name of the Son, so let's move on to the Holy Ghost:

But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you. (John 14:26 KJV)

Alright, so now that we know the name to baptize in, let's find out how:

Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. (Acts 2:38 KJV)

So, I think we can see now that baptism is absolutely essential to our salvation.

Communion

As far as communion goes, it's still a very necessary part of our salvation, and the process by which it should occur is pretty summed up by Paul. The emphasis is mine but it's an attempt to show what parts would need to occur in the service.

20 When ye come together therefore into one place, this is not to eat the Lord's supper. 21 For in eating every one taketh before other his own supper: and one is hungry, and another is drunken. 22 What? have ye not houses to eat and to drink in? or despise ye the church of God, and shame them that have not? What shall I say to you? shall I praise you in this? I praise you not. 23 For I have received of the Lord that which also I delivered unto you, That the Lord Jesus the same night in which he was betrayed took bread: 24 And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me. 25 After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me. 26 For as often as ye eat this bread, and drink this cup, ye do shew the Lord's death till he come. 27 Wherefore whosoever shall eat this bread, and drink this cup of the Lord, unworthily, shall be guilty of the body and blood of the Lord. 28 But let a man examine himself, and so let him eat of that bread, and drink of that cup. 29 For he that eateth and drinketh unworthily, eateth and drinketh damnation to himself, not discerning the Lord's body. 30 For this cause many are weak and sickly among you, and many sleep. 31 For if we would judge ourselves, we should not be judged. 32 But when we are judged, we are chastened of the Lord, that we should not be condemned with the world. (1 Corinthians 11:20-32 KJV)

One thing I'd like to point out as well:

...this is not to eat the Lord's supper

As you can see they were calling it the Lord's supper even then, and Paul let them know, that's not what this is.

I understand that Jesus fulfilled the law, but do these rituals as rituals inform our worship of God today?

Generally speaking, the act of worship today can be summed up pretty easily. In fact, you can see that worship began changing with David when he danced out of his clothing. Worship began to become more spontaneous. This would have never occurred with Moses or Aaron.

EDIT

Would you mind expanding your answer to include a discussion of this nature as a whole?

In regards to this, I believe that just about every ritual that existed in the old testament you will either find was fulfilled by Jesus (as shown with Aaron's sacrifice) or you will find that it still exists and that there is a prescribed way like with baptism and communion that still exists in the new testament.

However, you will find that there are ceremonies created by man which have confused the meaning of things. The one that comes to mind the quickest is marriage. Marriage is clearly defined in the bible as intercourse between a man and a woman and a ceremony or federal law have nothing to do with marriage in the eyes of God. But there is nothing wrong with the ceremony, I went through one with my wife :) ... you can find a more exhaustive explanation from my point of view on marriage here.