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From vCenter 5.0 you can go only up to 6.0 U2 VMware Product Interoperability Matrices
You should check this blog series as well before starting the upgrade The vSphere Upgrade Blog Series (Wrap-Up) - VMware vSphere Blog
Moderator: Moved to vSphere Upgade and Install
I had the same setup with my previous employer. Upgrading ESX 5.0 was not an option for us, due to the backup solution not supporting anything above 5.0.
What I ended up doing was creating a host with shared storage between vCenter 5.0 and vCenter 6.5. I created a powershell script to do the following;
1) Create/update backup
2) Migrate VM to migration host and LUN
3) Check all running services
4) Power off VM
5) Unregister VM in vCenter 5.0
6) Register VM in vCenter 6.5
7) Power on VM
8) Upgrade VM HW & Tools
9) Check running services
10) Migrate VM to correct 6.5 cluster
I found this the best way to do it, as there is always downtime involved in upgrading the whole stack.
Considering that you are running the hosts with following CPU series.
For PowerEdge R700 with Intel Xeon E5-2600 Series, the maximum supported version is ESXi 6.5 U3.
Related link - VMware Compatibility Guide - System Search
For PowerEdge R900 with Intel Xeon 73xx Series, the maximum supported version is ESXi 5.5 U3.
Note that the End of General Support for vSphere 5.5 was September 19, 2018.
Related link - VMware Knowledge Base
Technical Guidance for vSphere 5.5 is available until September 19, 2020 primarily through the self-help portal and telephone support is not provided.
Related link - VMware Product Lifecycle Policies
To answer to your questions:
Option 1 - We are planning to upgrade the ESXi 5.0 to 5.5 on vcenter 5.0u3 and then migrate to vcenter 6.5 to avoid license cost. Will it work ?
Option 2 - can we directly migrate ESX5.0 to vcenter6.5 and upgrade the ESX5.0 to 6.0 and then 6.5 ?
The vCenter always have to be on the higher build than the ESXi hosts.
So it is wise to upgrade the vCenter first and then the ESXi hosts.
The upgraded 6.5 vCenter can still manage the 5.5 hosts.
Note that only the R700 series supports upgrade till ESXi 6.5.
The following should be the viable plan for upgrading your environment:
Upgrade vCenter 5.0U3 --> 6.0U1 --> 6.5 U3
Upgrade R900 from ESXi 5.0 --> 5.5 U3 (out of support)
Upgrade R700 from ESXi 5.0 --> 6.0 U1 --> 6.5 U3
How to check supported upgrade paths and interoperability -
Check vCenter-ESXi version compatibility here - VMware Product Interoperability Matrices
vCenter upgrade path - VMware Product Interoperability Matrices
You can check ESXi upgrade path here - VMware Product Interoperability Matrices