After Brexit, would the UK be able to do trade deals with individual countries such as Germany, France, etc. without these deals having to be sanctioned by the European Union?
No, it won't be able to. The EU is the exclusive point of contact for trade deals with the EU and its members - or more precisely, for members of the European Union Customs Union.
Concluding international agreements on behalf of its members also is part of the EU's exclusive competences.
Edit to clarify: they will, of course, be able to negotiate trade deals with countries that are not part of the EU. Unless, that is, they remain within the European Union Customs Union.
No, as trade deals involving changing tariffs can be thought of as customs deals. Countries in the EU are in a customs union and have a common external tariff for goods, and so any changes to that tariff need to be made at the customs union level. Indeed, one of the main reasons given during the campaign for leaving the EU was the ability to conduct trade deals with non-EU countries which being in the EU didn't allow the UK to do. If the UK could sign a trade deal with Germany once the UK has left the EU, then it would stand to reason that Australia (to give an example) could sign a trade deal with the UK without the UK leaving the EU.
If countries in the EU customs union were able to make trade deals with third countries to import goods tariff free, then you would have issues where all goods moving from (say) Germany to France would have to be checked at the French border to check whether they originated in Germany and were therefore tariff free, or originated from a third country (say the UK post-Brexit) and therefore attracted a tariff for their movement into France. This would negate the entire point of the customs union, which is to remove those customs checks at borders.
Being a member of the single market is different and doesn't prevent a country signing trade deals with third countries (for example, Iceland and China recently signed a trade deal). Countries in the EEA are in the Single Market but not in the EU Customs Union (Switzerland is in a customs union with Liechtenstein but that is separate and goods moving from Germany to Switzerland go through normal customs procedures).