No, withdraw assets from the vault does not affect the reward boosting timer. Only when you claim your time is reset.
If I claim my xTETU rewards before 28 days I forfeit a portion of them?
Yes, the reward boosting system's locking period is 28 days so there is no penalty on rewards.
Can I withdraw my assets from the vault without claim rewards and then claim rewards after the 28 day period?
Yes, if you do this you will not lose any rewards because the timer will be full after 28 days.
If I have the full timer and withdraw and deposit back into the vault is my timer reset?
No, Only when you claim your time is reset.
Are vault rewards fixed? If you double the TVL the rewards will halve?
Rewards are automatically distributed according to the profitability of the vault. So if there is a big increase in TVL the increase in profit will readjust the distribution of rewards, this will offset the dilution of rewards.