Could I ask to start a bit later?
Yes, no problem with that. Be precise though, tell them which start date would work the best for you. If they accept to let you start later, be sure you can make it work, because if you ask them twice it won't be appreciated at all.
how would the company perceive if I ask to work remotely and travel to work on site a few days a week?
Do you know if they let other employees work remotely ? If not, no, they won't want you to either, so I wouldn't bother asking.
Interns need more supervision and care in general. That's why teleworking is rare for interns (that I know). You're new to the work world, you need to be managed/trained/supervised more closely. Even for regular employees, if they start at a company with the expectation of working remotely, they often don't telework the first month or so, because there's so much to learn it's better to be at the office for a while. And as an intern, you also don't have a record of working well with other companies, or demonstrate a strong work ethic, so asking to telework in those conditions is a huge ask.
So if you really want to ask them, frame it as such. Don't just mention it as something casual, tell them you know you're possibly inconveniencing them.
Another thing : I disagree with "there's no harm in asking". You're already asking to start later. It would be a red flag to me that after you accepted a position, you seem to have all these problems to be able to work with us. Didn't you do your research before ? Weren't you enough prepared ? Did you know you would have all these problems and waited for the last minute to ask those things from us ? Was this your plan all along, or didn't you think things through ? Both of these things aren't good. They probably won't pull the internship because of this, but there's a good chance the good impression they had of you will degrade. You'll have to work harder to make up for it.
Or not. It does depend on the company and the manager, but I don't want you to think there are no possible downsizes to asking them something that's out of the norm and in combination with asking them another favor. So tread lightly, and ask them only of not working remotely is really important to you.
But I want you to think also about your own experience. Working remotely might not be best for you. Part of an internship is to learn how to work in an office setting. This is valuable experience. You'll learn a lot about working with others, handle interpersonal problems, adapt to a work culture...