If UK stays in the customs union does that mean that freedom of relocation (within the EU) will be maintained?

No deal so it looks like nothing is going to go in or out of the UK. They have royally screwed themselves. Nobody could have done it worse. Even Hitler could not have come up with such a scheme to destroy the UK.

no. freedom of movement and customs union are NOT the same thing.

A straightforward customs union would not provide for that. Turkey, for example, is in a customs with the EU and its citizens don't have freedom of movement. Membership of the European Free Trade Association/European Economic Area would provide for free movement but not so free as EU membership. Liechtenstein for example is an EFTA member but has quotas for the entry of EU nationals.

Unknowable but unlikely. The idea that the UK can keep all the benefits of EU membership with none of the downsides isn't playing very well with Brussels. However, it's possible that Brit expats already settled in the EU would be grandfathered in and not have to uproot themselves.

I mean freedom of relocation both into UK from the EU and from UK to the EU nations

Depends on the rest of the deal

It's however unlikely that freedom of movement will be retained if the UK does leave the EU

This would come from EEA. But yes, the EEA (and the EFTA) would likely be included in any customs union deal.