I will try my best to answer your questions.
1) So what I understand is , If we are playing in Key of CMajor , we can play only these chords with Whatever Roman Numeral Chord Progression. And if we have to shift the key, we have to go according to circle of Fifth and that will be Gmajor Key next. Is that correct ?
A1) To your first question/statement "If we are playing in Key of CMajor , we can play only these chords with Whatever Roman Numeral Chord Progression" I am not sure what to say. It is somewhat true in the sense that if you play something not in the list you will have "accidentals" that are NOT in key. However, this is not forbidden. You are not required to stick only to these chords. But deviations from them have to smoothly voice lead back into the key. Here are a couple counter examples. In Rhythm Changes the basic chords are I --> VI7 --> ii-7 --> V7. Notice that the 6 chord which is usually minor is major here. It provides a leading tone to the ii-7. You could say that the key changed momentarily or that one should treat the key as minor in the ii-7 but you don't need to. In fact many songs based on these changes do not have an out of key note in the melody yet use the VI7 instead of the vi-7. This brings me to my second counter example. One frequently uses cycle extensions to "lead into" chords that are in key. And example might be I --> iii-7 --> III7 --> vi-7 --> VI7 --> ii-7 --> V7 --> I. The iii and the vi chords are played "in key" then changed to be a dom 7th for the next chord in sequence until you get to the classic ii-->V-->I. You can apply this as often or as little as desired. Formally speaking you might say the key changed with every occurrence of this device. You can also think of these as "passing chords" containing accidentals. As an aside chord progression and song don't always belong to a fixed key. We sometimes vamp back and forth between chords that sound good together but have no relation based on key signature or analysis using a fixed key. As for your second statement in this question "And if we have to shift the key, we have to go according to circle of Fifth and that will be Gmajor Key next" the answer is NO. You do not have to change key by a 5th. In fact the most common device I know going back to before classical music is to change to the 4th. For example from the key of C to the key of F. In fact you can change key from any starting point to any other key! There is a great book called Modulations by Reger that outlines just about every possible key change with a smooth chord progression. This is quite advanced.
2) If we are playing in Key of C Major and lets say a solo comes , so the solo will be on a C Major scale or its relative minor A minor Scale but not any other scale?
A2) For this most part this is a safe bet, a good starting point. You can move away from the key in your solo but you'd be wise to respect the structure of the song. Keys come in comparable sets that share a tetra chord, no surprise these are related by a 5th. For example the keys of F, C, and G (the IV, I, and V in the key of C) all overlap. This means you can deviate from one to the other and com back smoothly. Going much further from the key in many cases licks based on the minor blues scale sound fine even in a major key. This is a common occurrence in Rock and Jazz. So in fact you don't need to stay in key or on the major scale of the key but your musical ideas should respect the key. As you learn more you will come across all sorts of formulaic methods for soloing. Not worth discussing here since you are just starting.
3) If we are playing in a key, we still play the Open chords in a guitar on first 4 frets , then what’s the point of learning all the Patterns and creating chords on lets say Fifth Fret with Amajor Scale ? Are those patterns only for playing Lead ? And we can play simple open chords for Rhythm?
A3) This is a different question altogether. Everything is important. The open string chord patterns are very important but so are the movable ones. They are not just for soloing. You cannot play all 12 keys with open string chord voicings only, and part of the value of the other chord patterns is the alternate voicing (different order of notes create different harmony). You definitely want to learn all you can here to be a great rhythm player.
4) When we say We play in a key , that means we can freely use the open chords? Even if we are playing In key of A Major whose pattern 1 root starts on fifth fret of guitar ?
A4) This question is not coherent or clear to me. I cannot understand what you are asking. First of all there are keys for which NONE OF THE OPEN CHORDS WORK. So your first statement is not true. I am also not quite sure what your second statement means. The pattern of A major starts on the fifth fret. This is not true, unless you are referring to a specific method book and I don't know that notation. The A major scale starts on the open A string in most books. But if you are referring to the standard major scale starting in the second finger on the low E string then, yes you can still play the open A chord as rhythm. There is no reason why you need to be in the same position when soloing as you are when playing rhythm. In fact there are several ways to play each chord and each has a corresponding scale pattern in the same position. This is sometimes referred to as the C-A-G-E-D system.