1. The residue in a particular chain does not always start with index 1.
2. There are some gaps in residue sequences for example 50-58. There is a jump from 50 to 58. There are no residue in between.
3. There are sometimes residue length dont match with the given FASTA sequence."

Welcome to structural biology!

Unless I am misunderstanding this is all common. (I'll get back to that.).
One thing to make your post more clear and better able to be answered is to point to examples. For example you bring up, "for example 50-58. There is a jump from 50 to 58." Which PDB id code are you referring to here? A specific one? A lot? Cite the PDB id records if it is just one or a few. They may all be from the same protein?

And that citing becomes more important in the next section. I suspect it is a simple explanation but we cannot respond well without examples.

Back to what is going on here in the three bullet points you posted...

Please read the Proteopedia entry 'Unusual sequence numbering'. That should definitely cover bullet points #1 and #2. In particular the 'Gaps In Sequence Numbering' section addresses number 2.
Without examples, I am having trouble getting at what you mean by bullet point #3 there. Is it PDB differing from Uniprot? Then see 'Renumbering PDB files' under 'See Also' section there. Or something else? It used to be the PDB pages were poor for relating the missing residues, and you'd need to check out specifically the information at PDBsum. (You can see more about this here with an example with images.) There was a similar thing about the FASTA files, too. The ones at PDB would show the sequence of what the scientist used in construction, and the PDBsum resource would give you a FASTA sequence including only residues represented in the resulting structure. And so I'm wondering if you are asking about that?

• The residue numbering refers to the protein that was used for crystallization. Meaning, that was the intended sequence of a protein that was cloned and purified, and eventually used for structure determination. It often happens that residues at either end are not visible. There are at least two reasons for it: those residues are flexible (don't have a single conformation in crystals) and their diffraction pattern is not strong enough to fit anything into the map. The other reason is that parts of the protein may get degraded during purification or crystallization. Either way, those residues can't be modelled, and are omitted from the PDB file.
• Same answer as above, except that flexible residues can occur anywhere in the chain.
• Same answer as above, depending on where you got the FASTA sequences. PDB sequence records usually contain the expected protein sequence. If you look for the word missing in PDB files, it will indicate when some residues are absent with regard to that expectation.

The main point is that everything you observed is common and normal, even though it may have startled you if this is the first time you looked at many PDB structures at once.