Mdthbs

Assume a Divine Soul Sorcerer. The spellcasting feature of the class states (highlight mine):

Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.

Can I do this when I gain level 1 in the class?

Let's say I choose the affinity good, which gives me the cleric spell "Cure Wounds" because of the Divine Magic feature:

Your link to the divine allows you to learn spells from the cleric class. When your Spellcasting feature lets you learn or replace a sorcerer cantrip or a sorcerer spell of 1st level or higher, you can choose the new spell from the cleric spell list or the sorcerer spell list. You must otherwise obey all the restrictions for selecting the spell, and it becomes a sorcerer spell for you.

In addition, choose an affinity for the source of your divine power: good, evil, law, chaos, or neutrality. You learn an additional spell based on that affinity, as shown below. It is a sorcerer spell for you, but it doesn't count against your number of sorcerer spells known. If you later replace this spell, you must replace it with a spell from the cleric spell list.

If my class allows me to replace a spell when I gain Level 1, then the Divine Magic would allow me to replace the "Cure Wounds" with for example "Guiding Bolt".

Now intuitively I would say no, because Level 1 is the first level and you can't have less, so you don't "gain" it. However some features explicitly state "Starting at 1st level", for example the feature "Favored by the Gods" from the same Divine Soul Sorcerer. This does hint to me that there might be a "gaining" of 1st level, otherwise this extra wouldn't be necessary.

Starting at 1st level, divine power guards your destiny. If you fail a saving throw or miss with an attack roll, you can roll 2d4 and add it to the total, possibly changing the outcome. Once you use this feature, you can't use it again until you finish a short or long rest.

Bonus question: Is it any different if I multiclass into sorcerer and my first Divine Soul Sorcerer level is not my first character level?

You don't "gain" your first character level

The rules for creating characters define what "gaining a level" means in the beyond first level section:

Beyond 1st Level

As your character goes on adventures and overcomes challenges, he or she gains experience, represented by experience points. A character who reaches a specified experience point total advances in capability. This advancement is called gaining a level.

The entire section repeatedly uses the wording "when you gain a level" for things that you obtain on levels beyond the first.

An example of another class that gets features when they "gain a level" is the wizard's spellcasting section which states:

Learning Spells of 1st Level and Higher

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free.

If you also considered that you "gained" your first wizard level, you would then add an additional 2 spells to your spellbook for free beyond the 6 your spellbook starts with. I have never seen anyone consider that this is the case and the pregen wizard character only starts with 6 1st level spells which supports the interpretation that the first character level is not really "gained".

Therefore, the easier interpretation is that your features that state "when you gain a level" in general should apply to getting levels beyond first.

Multiclassing is difficult

You also stated that you were interested in knowing if multiclassing would change this. Indeed, the rules for multiclassing do state that you are "gaining a first level in a class" in the section introduction:

With this rule, you have the option of gaining a level in a new class whenever you advance in level, instead of gaining a level in your current class.

And when mentioning gained proficiencies:

When you gain your first level in a class other than your initial class, you gain only some of new class's starting proficiencies, as shown in the Multiclassing Proficiencies table.

It is also no longer your first character level so the reasoning used above would not apply. This would allow the interpretation that you are indeed "gaining a sorcerer level", therefore letting you retrain the spell as you wanted.

That said, though, considering multiclassing is already an optional rule that you need explicit DM permission to use, asking your DM about whether multiclassing changes this for a Divine Soul Sorcerer would be part of the conversation to have with your DM.

Personal Opinion

As a DM, I'd not allow multiclassing to change whether you can retrain the first level spell. If a player really wanted to get the extra retraining on sorcerer level 1, I might just allow it for a single classed character (though note how that makes the whole thing about choosing the alignment completely pointless, you might as well just gain any 1st level cleric spell) or just not allow it at all even with multiclassing. (Similarly, I'd also not have a multiclassed wizard start with 8 spells, that's just weird to me).