Translation:I swim a lot in the summer.
Zeltefka, it doesn't say natsutoki. That's just not a construction that would be used. If you wanted to specifically say that you swam a lot in Summer then you might say natsu ni (in summer), or you could also say natsu toki ni (when it is summer) - although you'd probably either want to have no or de aru between natsu and toki ni. Natsu wa is more a general way of saying that you're doing something when it's summer, like saying generally, I swim a lot in Summer as opposed to I swim a lot specifically in summer (if ni followed natsu instead of wa). Make sense?
Different construction for during/while. That would use 間 (aida) or 内 (uchi) like as follows.
The difference between the 2 is that 間 marks it as a time or duration while 内 refers to the inside period within a specific instance.
As explained elsewhere, 夏の時に would be roughly mean "when it is summertime..."
I would prefer to use either of the two constructions here over what they have written though...
Because ni is more specific. If you use ni you're saying that you swim a lot specifically in summer. You wouldn't use de. De is used to mean by to show the means of transport you take eg. Basu de gakkou ni ikimasu - I go to school By bus. It can mean with when you use a tool of some kind eg hashi de tabemasu - I eat WITH chopsticks. And it can mean at to indicate you're doing something at a specific location eg. Gakkou de manabimasu - I learn AT school.
たくさん is a quantitative adjective, it is not describing the verb - it is describing the amount of swimming being done. よくis an adverb and it describes the frequency of the action being performed. So if you were to use it in this sentence instead of たくさん, it would mean I swim often in summer.
No it's not the object. It's describing the amount of swimming done - as opposed to the frequency of swimming. There IS no object in this sentence. In the sentence I eat cake - cake is the object - it is what is being eaten. In the sentence I buy a dog - the dog is the object - it is what is being bought. Even if you said I swim in the pool there is still no object - the pool is not the object of the verb/swim - it is merely the location where you are swimming.
No, it's an adverbial (of degree, I think, telling how much the action is done). The verb "swim" doesn't take an object -- you don't "swim something". Well, you could swim a lap, or a race, but generally you just swim. This is called an intransitive verb, as opposed to transitive verbs which have objects.
No. を indicates that the word directly proceeding it is the direct object of the verb or in other words it is object of the action eg. I ate cake - I is the subject - the person performing the action/the eating, the cake is what I ate so it is the object. In this original sentence - I swim a lot in summer - there is no object. By putting たくさん before を you are saying that it is the direct object of the verb which is just grammatically impossible.
たくさん is a quantitative adjective - it describes the amount of swimming. あんまり is an adverb, but a very distinctive and special adverb in that it is ONLY ever used with a negative verb and it describes the negative frequency with which an action is carried out eg. ケーキをあんまり食べません I don't really or I rarely or I hardly ever eat cake.