Sizing Stories in a Cross-Functioan Agile Team

Short answer is Yes and Yes.

The goal of estimation is to estimate the entire effort to accomplish the goal of the story in relation to other items in the backlog that are yet to be started.  Velocity is a measure of how much work a team can do during a specific time period.

Teams are the focus of all agile practices.  Individual effort should always be targeted towards helping the team achieve their goals. 

Parting comment.  Estimates are guesses. Do not feel like it has to be perfect and I wouldn't do any kind of formula (backend estimate + frontend estimate + QC estimate).  The team needs to discuss and determine an estimate as a complete team.  Since you mentioned story points I will add that they are not individul estimates.  They are estimates of effort as related to other items that are yet to be done.  

Should we get one size per each story? and calculate one velocity across all team members?

Anything else would suggest a problem. The role is Development Team for a reason: team members are jointly responsible for the work done and for any estimates that are made.

Remember that velocity ought to be calculated for work which the team actually completes.

Just an additional thought, that may help your cross-functional team adapt to one (and only one) size estimate for an item to meet DoD.

Most teams estimate in a range across 5 different values.   If using Fibonacci, that usually results in sizing ranges of 1,2,3,5,8, or 3,5,8,13,21, etc,

You can compare such range values to t-shirt sizes, and simply explain to the Development Team that the objective is to place each item into either a very small, small, medium, large, or extra-large bucket.   As a cross-functional Development Team, they should be able to discuss and roughly size each item into one of those size buckets.

Hi everyone, 

How should the estimation work for a larger project? We have 7 scrum teams mixed of FE and BE devs and too many issues during the release planning as it's hard to distribute epics to teams based on FE/BE capacity. 

Many thanks. 

Well, first of all stop distributing work to teams. Encourage them to self-manage, and to pull work when they have the capacity and focus to finish it to a Done standard without "issues". Only the people doing the work can really estimate this.